| L(s) = 1 | − 10·7-s − 4·9-s + 16·17-s − 8·23-s − 10·25-s + 8·31-s + 20·41-s + 24·47-s + 55·49-s + 40·63-s + 48·73-s − 40·79-s + 7·81-s + 16·89-s + 32·97-s + 24·103-s + 12·113-s − 160·119-s − 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 64·153-s + 157-s + ⋯ |
| L(s) = 1 | − 3.77·7-s − 4/3·9-s + 3.88·17-s − 1.66·23-s − 2·25-s + 1.43·31-s + 3.12·41-s + 3.50·47-s + 55/7·49-s + 5.03·63-s + 5.61·73-s − 4.50·79-s + 7/9·81-s + 1.69·89-s + 3.24·97-s + 2.36·103-s + 1.12·113-s − 14.6·119-s − 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 5.17·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{90} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{90} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.201088112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.201088112\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + T )^{10} \) |
| good | 3 | \( 1 + 4 T^{2} + p^{2} T^{4} + 64 T^{6} + 226 T^{8} + 152 p T^{10} + 226 p^{2} T^{12} + 64 p^{4} T^{14} + p^{8} T^{16} + 4 p^{8} T^{18} + p^{10} T^{20} \) |
| 5 | \( 1 + 2 p T^{2} + 57 T^{4} + 288 T^{6} + 282 p T^{8} + 7188 T^{10} + 282 p^{3} T^{12} + 288 p^{4} T^{14} + 57 p^{6} T^{16} + 2 p^{9} T^{18} + p^{10} T^{20} \) |
| 11 | \( 1 + 12 T^{2} + 469 T^{4} + 4592 T^{6} + 97954 T^{8} + 769928 T^{10} + 97954 p^{2} T^{12} + 4592 p^{4} T^{14} + 469 p^{6} T^{16} + 12 p^{8} T^{18} + p^{10} T^{20} \) |
| 13 | \( 1 + 2 p T^{2} + 329 T^{4} + 7520 T^{6} + 122050 T^{8} + 1329972 T^{10} + 122050 p^{2} T^{12} + 7520 p^{4} T^{14} + 329 p^{6} T^{16} + 2 p^{9} T^{18} + p^{10} T^{20} \) |
| 17 | \( ( 1 - 8 T + 61 T^{2} - 288 T^{3} + 1586 T^{4} - 6320 T^{5} + 1586 p T^{6} - 288 p^{2} T^{7} + 61 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 19 | \( 1 + 68 T^{2} + 2729 T^{4} + 83072 T^{6} + 2077858 T^{8} + 43639752 T^{10} + 2077858 p^{2} T^{12} + 83072 p^{4} T^{14} + 2729 p^{6} T^{16} + 68 p^{8} T^{18} + p^{10} T^{20} \) |
| 23 | \( ( 1 + 4 T + 55 T^{2} + 256 T^{3} + 78 p T^{4} + 8840 T^{5} + 78 p^{2} T^{6} + 256 p^{2} T^{7} + 55 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 29 | \( 1 + 106 T^{2} + 6453 T^{4} + 312056 T^{6} + 12090386 T^{8} + 382188924 T^{10} + 12090386 p^{2} T^{12} + 312056 p^{4} T^{14} + 6453 p^{6} T^{16} + 106 p^{8} T^{18} + p^{10} T^{20} \) |
| 31 | \( ( 1 - 4 T + 59 T^{2} + 144 T^{3} - 246 T^{4} + 17000 T^{5} - 246 p T^{6} + 144 p^{2} T^{7} + 59 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 37 | \( 1 + 154 T^{2} + 13317 T^{4} + 812728 T^{6} + 38862418 T^{8} + 1559615004 T^{10} + 38862418 p^{2} T^{12} + 812728 p^{4} T^{14} + 13317 p^{6} T^{16} + 154 p^{8} T^{18} + p^{10} T^{20} \) |
| 41 | \( ( 1 - 10 T + 85 T^{2} - 568 T^{3} + 5682 T^{4} - 41084 T^{5} + 5682 p T^{6} - 568 p^{2} T^{7} + 85 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 43 | \( 1 + 156 T^{2} + 14933 T^{4} + 1045808 T^{6} + 58136290 T^{8} + 2734211560 T^{10} + 58136290 p^{2} T^{12} + 1045808 p^{4} T^{14} + 14933 p^{6} T^{16} + 156 p^{8} T^{18} + p^{10} T^{20} \) |
| 47 | \( ( 1 - 12 T + 187 T^{2} - 1488 T^{3} + 15610 T^{4} - 97224 T^{5} + 15610 p T^{6} - 1488 p^{2} T^{7} + 187 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 53 | \( 1 + 250 T^{2} + 29925 T^{4} + 2489016 T^{6} + 172171026 T^{8} + 189858924 p T^{10} + 172171026 p^{2} T^{12} + 2489016 p^{4} T^{14} + 29925 p^{6} T^{16} + 250 p^{8} T^{18} + p^{10} T^{20} \) |
| 59 | \( 1 + 388 T^{2} + 67929 T^{4} + 7258112 T^{6} + 556921826 T^{8} + 35114903880 T^{10} + 556921826 p^{2} T^{12} + 7258112 p^{4} T^{14} + 67929 p^{6} T^{16} + 388 p^{8} T^{18} + p^{10} T^{20} \) |
| 61 | \( 1 + 410 T^{2} + 82985 T^{4} + 10871968 T^{6} + 1019495618 T^{8} + 71469360820 T^{10} + 1019495618 p^{2} T^{12} + 10871968 p^{4} T^{14} + 82985 p^{6} T^{16} + 410 p^{8} T^{18} + p^{10} T^{20} \) |
| 67 | \( 1 + 476 T^{2} + 109925 T^{4} + 16248496 T^{6} + 1701824546 T^{8} + 131801966440 T^{10} + 1701824546 p^{2} T^{12} + 16248496 p^{4} T^{14} + 109925 p^{6} T^{16} + 476 p^{8} T^{18} + p^{10} T^{20} \) |
| 71 | \( ( 1 + 235 T^{2} - 416 T^{3} + 24994 T^{4} - 57920 T^{5} + 24994 p T^{6} - 416 p^{2} T^{7} + 235 p^{3} T^{8} + p^{5} T^{10} )^{2} \) |
| 73 | \( ( 1 - 24 T + 469 T^{2} - 6080 T^{3} + 69298 T^{4} - 621392 T^{5} + 69298 p T^{6} - 6080 p^{2} T^{7} + 469 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 79 | \( ( 1 + 20 T + 435 T^{2} + 5104 T^{3} + 64226 T^{4} + 545208 T^{5} + 64226 p T^{6} + 5104 p^{2} T^{7} + 435 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 83 | \( 1 + 660 T^{2} + 207145 T^{4} + 40565696 T^{6} + 5478581410 T^{8} + 532641853160 T^{10} + 5478581410 p^{2} T^{12} + 40565696 p^{4} T^{14} + 207145 p^{6} T^{16} + 660 p^{8} T^{18} + p^{10} T^{20} \) |
| 89 | \( ( 1 - 8 T + 373 T^{2} - 2560 T^{3} + 62050 T^{4} - 326640 T^{5} + 62050 p T^{6} - 2560 p^{2} T^{7} + 373 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 97 | \( ( 1 - 16 T + 221 T^{2} - 1920 T^{3} + 10914 T^{4} - 72928 T^{5} + 10914 p T^{6} - 1920 p^{2} T^{7} + 221 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.93000691842654722395498413631, −2.90595364755565441750168195527, −2.89841585752382476635408180733, −2.74288310478247974909740217160, −2.64846360877078120164332261735, −2.57591563725365522360178328446, −2.38699637504592061474351877144, −2.26620800522007891059416323804, −2.22111149169887427305105749165, −2.12586740472679978355687509686, −1.98535265001311835568603038253, −1.94701771383973855384538746851, −1.82319913526552768991724787840, −1.81319130121992479285080289230, −1.47087395195713120301872347826, −1.29927155553044815731985462097, −1.18568977104852461809194157782, −1.04648562984296248505563359556, −0.824086865218739138294909849540, −0.800771961243541812890219023487, −0.78199942151212184637082963039, −0.64117596722209019796145249257, −0.50491078056327215459534228135, −0.32050242043429450855035820356, −0.10474674767365683762810363620,
0.10474674767365683762810363620, 0.32050242043429450855035820356, 0.50491078056327215459534228135, 0.64117596722209019796145249257, 0.78199942151212184637082963039, 0.800771961243541812890219023487, 0.824086865218739138294909849540, 1.04648562984296248505563359556, 1.18568977104852461809194157782, 1.29927155553044815731985462097, 1.47087395195713120301872347826, 1.81319130121992479285080289230, 1.82319913526552768991724787840, 1.94701771383973855384538746851, 1.98535265001311835568603038253, 2.12586740472679978355687509686, 2.22111149169887427305105749165, 2.26620800522007891059416323804, 2.38699637504592061474351877144, 2.57591563725365522360178328446, 2.64846360877078120164332261735, 2.74288310478247974909740217160, 2.89841585752382476635408180733, 2.90595364755565441750168195527, 2.93000691842654722395498413631
Plot not available for L-functions of degree greater than 10.
