| L(s) = 1 | + 2·3-s − 3·4-s + 5·9-s − 6·12-s − 10·13-s + 6·16-s − 4·17-s − 6·23-s + 6·25-s − 6·27-s + 16·29-s − 15·36-s − 20·39-s + 16·43-s + 12·48-s − 3·49-s − 8·51-s + 30·52-s − 36·53-s + 10·61-s − 10·64-s + 12·68-s − 12·69-s + 12·75-s − 14·79-s − 22·81-s + 32·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3/2·4-s + 5/3·9-s − 1.73·12-s − 2.77·13-s + 3/2·16-s − 0.970·17-s − 1.25·23-s + 6/5·25-s − 1.15·27-s + 2.97·29-s − 5/2·36-s − 3.20·39-s + 2.43·43-s + 1.73·48-s − 3/7·49-s − 1.12·51-s + 4.16·52-s − 4.94·53-s + 1.28·61-s − 5/4·64-s + 1.45·68-s − 1.44·69-s + 1.38·75-s − 1.57·79-s − 2.44·81-s + 3.43·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9541284696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9541284696\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 7 | \( ( 1 + T^{2} )^{3} \) |
| 13 | \( 1 + 10 T + 61 T^{2} + 256 T^{3} + 61 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 3 | \( ( 1 - T - T^{2} + 8 T^{3} - p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 5 | \( 1 - 6 T^{2} + 11 T^{4} - 44 T^{6} + 11 p^{2} T^{8} - 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - T^{2} + 251 T^{4} - 18 p T^{6} + 251 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 2 T + 7 T^{2} + 12 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 86 T^{2} + 3451 T^{4} - 82508 T^{6} + 3451 p^{2} T^{8} - 86 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 3 T + 53 T^{2} + 106 T^{3} + 53 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 8 T + 67 T^{2} - 288 T^{3} + 67 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 121 T^{2} + 7651 T^{4} - 293638 T^{6} + 7651 p^{2} T^{8} - 121 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 157 T^{2} + 11771 T^{4} - 540558 T^{6} + 11771 p^{2} T^{8} - 157 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 133 T^{2} + 8499 T^{4} - 385678 T^{6} + 8499 p^{2} T^{8} - 133 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 8 T + 105 T^{2} - 560 T^{3} + 105 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 137 T^{2} + 8067 T^{4} - 352934 T^{6} + 8067 p^{2} T^{8} - 137 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 6 T + p T^{2} )^{6} \) |
| 59 | \( 1 - 66 T^{2} + 10763 T^{4} - 439844 T^{6} + 10763 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 - 5 T + 181 T^{2} - 608 T^{3} + 181 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 289 T^{2} + 39707 T^{4} - 3327366 T^{6} + 39707 p^{2} T^{8} - 289 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 90 T^{2} + 12959 T^{4} - 845804 T^{6} + 12959 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 349 T^{2} + 54115 T^{4} - 4959166 T^{6} + 54115 p^{2} T^{8} - 349 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 7 T + 149 T^{2} + 1254 T^{3} + 149 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 146 T^{2} + 27387 T^{4} - 2110724 T^{6} + 27387 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 218 T^{2} + 26655 T^{4} - 2647148 T^{6} + 26655 p^{2} T^{8} - 218 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 397 T^{2} + 77507 T^{4} - 9352542 T^{6} + 77507 p^{2} T^{8} - 397 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.98439109258110554584549561599, −6.80358638778763450673321353460, −6.79232533876275435002204542431, −6.46904732687833584282634425541, −6.39132692428128971315530933615, −5.95069605969590117723883622220, −5.78873243066428453348730183154, −5.51621091837391478117587636996, −5.40075158968752373583365387633, −5.00254300325138673382042963811, −4.84047945375030087270046649682, −4.65569962924037068855490432691, −4.44185674711972895320346406952, −4.44158423768658849348999576109, −4.05953292986662558846931400662, −3.97414750535865299710012834716, −3.68606491323317005485094810225, −3.13493755301173827709043128841, −2.92365771622626610401191388108, −2.86526960208275699524225764029, −2.55841522382452795990575021666, −2.09140185575215345124592208722, −1.78688339710042113122248251601, −1.47175011651461004066187776997, −0.48312153935958298843761971831,
0.48312153935958298843761971831, 1.47175011651461004066187776997, 1.78688339710042113122248251601, 2.09140185575215345124592208722, 2.55841522382452795990575021666, 2.86526960208275699524225764029, 2.92365771622626610401191388108, 3.13493755301173827709043128841, 3.68606491323317005485094810225, 3.97414750535865299710012834716, 4.05953292986662558846931400662, 4.44158423768658849348999576109, 4.44185674711972895320346406952, 4.65569962924037068855490432691, 4.84047945375030087270046649682, 5.00254300325138673382042963811, 5.40075158968752373583365387633, 5.51621091837391478117587636996, 5.78873243066428453348730183154, 5.95069605969590117723883622220, 6.39132692428128971315530933615, 6.46904732687833584282634425541, 6.79232533876275435002204542431, 6.80358638778763450673321353460, 6.98439109258110554584549561599
Plot not available for L-functions of degree greater than 10.
