| L(s) = 1 | − 12·3-s − 12·5-s + 44·7-s + 72·9-s − 296·11-s + 316·13-s + 144·15-s − 220·17-s − 528·21-s + 652·23-s + 714·25-s − 1.18e3·27-s − 2.76e3·31-s + 3.55e3·33-s − 528·35-s − 2.05e3·37-s − 3.79e3·39-s − 4.31e3·41-s − 1.72e3·43-s − 864·45-s − 5.25e3·47-s + 968·49-s + 2.64e3·51-s + 6.92e3·53-s + 3.55e3·55-s − 1.21e4·61-s + 3.16e3·63-s + ⋯ |
| L(s) = 1 | − 4/3·3-s − 0.479·5-s + 0.897·7-s + 8/9·9-s − 2.44·11-s + 1.86·13-s + 0.639·15-s − 0.761·17-s − 1.19·21-s + 1.23·23-s + 1.14·25-s − 1.62·27-s − 2.87·31-s + 3.26·33-s − 0.431·35-s − 1.49·37-s − 2.49·39-s − 2.56·41-s − 0.932·43-s − 0.426·45-s − 2.37·47-s + 0.403·49-s + 1.01·51-s + 2.46·53-s + 1.17·55-s − 3.25·61-s + 0.798·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.06173348397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06173348397\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 12 T - 114 p T^{2} + 12 p^{4} T^{3} + p^{8} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + 2 p T + 2 p^{2} T^{2} + 2 p^{5} T^{3} + p^{8} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 44 T + 968 T^{2} + 19844 p T^{3} - 223634 p^{2} T^{4} + 19844 p^{5} T^{5} + 968 p^{8} T^{6} - 44 p^{12} T^{7} + p^{16} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 148 T + 23158 T^{2} + 148 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 316 T + 49928 T^{2} - 856676 p T^{3} + 14434846 p^{2} T^{4} - 856676 p^{5} T^{5} + 49928 p^{8} T^{6} - 316 p^{12} T^{7} + p^{16} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 220 T + 24200 T^{2} - 710380 T^{3} - 7504709618 T^{4} - 710380 p^{4} T^{5} + 24200 p^{8} T^{6} + 220 p^{12} T^{7} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 427396 T^{2} + 77993999686 T^{4} - 427396 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 652 T + 212552 T^{2} - 183067908 T^{3} + 157672813838 T^{4} - 183067908 p^{4} T^{5} + 212552 p^{8} T^{6} - 652 p^{12} T^{7} + p^{16} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 851332 T^{2} + 370965009478 T^{4} - 851332 p^{8} T^{6} + p^{16} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 1380 T + 2033142 T^{2} + 1380 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 2052 T + 2105352 T^{2} + 1486261548 T^{3} - 272964536242 T^{4} + 1486261548 p^{4} T^{5} + 2105352 p^{8} T^{6} + 2052 p^{12} T^{7} + p^{16} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 2156 T + 3055206 T^{2} + 2156 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 1724 T + 1486088 T^{2} + 1694904052 T^{3} - 3999465653426 T^{4} + 1694904052 p^{4} T^{5} + 1486088 p^{8} T^{6} + 1724 p^{12} T^{7} + p^{16} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 5252 T + 13791752 T^{2} + 34933252588 T^{3} + 85343693886478 T^{4} + 34933252588 p^{4} T^{5} + 13791752 p^{8} T^{6} + 5252 p^{12} T^{7} + p^{16} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6924 T + 23970888 T^{2} - 87091477572 T^{3} + 294447643869134 T^{4} - 87091477572 p^{4} T^{5} + 23970888 p^{8} T^{6} - 6924 p^{12} T^{7} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 16072452 T^{2} + 338035836658118 T^{4} - 16072452 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 6060 T + 23462982 T^{2} + 6060 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 6012 T + 18072072 T^{2} + 131433977268 T^{3} + 952961855450318 T^{4} + 131433977268 p^{4} T^{5} + 18072072 p^{8} T^{6} + 6012 p^{12} T^{7} + p^{16} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 1116 T + 35254326 T^{2} - 1116 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 244 p T + 29768 p^{2} T^{2} - 15649149596 p T^{3} + 6949639479021838 T^{4} - 15649149596 p^{5} T^{5} + 29768 p^{10} T^{6} - 244 p^{13} T^{7} + p^{16} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 60033796 T^{2} + 2261038014449926 T^{4} - 60033796 p^{8} T^{6} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16332 T + 133367112 T^{2} + 713383376868 T^{3} + 3801627135818318 T^{4} + 713383376868 p^{4} T^{5} + 133367112 p^{8} T^{6} + 16332 p^{12} T^{7} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 156089476 T^{2} + 13455945186102406 T^{4} - 156089476 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 11812 T + 69761672 T^{2} - 1162925642988 T^{3} + 19287468448798478 T^{4} - 1162925642988 p^{4} T^{5} + 69761672 p^{8} T^{6} - 11812 p^{12} T^{7} + p^{16} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.35561803397674411099398316399, −10.79385868558341383901081913603, −10.75797221882905724367195367605, −10.74467976594632696587611396444, −10.33476274392271626657055836712, −9.806615495525444337359550466908, −9.141931480489875094216659074786, −9.077800964433028957343356334296, −8.378993031342201575205071941172, −8.221926532255516463626822504672, −8.145413864702020628307011276755, −7.30230905447808388992626924352, −7.06822993089736014769140693933, −6.93762712238470241735905450364, −6.22141577204021419068447328967, −5.74324186231419842167196673709, −5.36999767415097759849494969725, −5.08147379778681274403972326026, −4.90734721876261123874445241170, −4.16454323899122216656106174179, −3.39775794188556722192905451134, −3.19811716933352371559581142205, −1.94705735032545744072602092616, −1.47994745097478402845102598562, −0.10183671525259519435567250948,
0.10183671525259519435567250948, 1.47994745097478402845102598562, 1.94705735032545744072602092616, 3.19811716933352371559581142205, 3.39775794188556722192905451134, 4.16454323899122216656106174179, 4.90734721876261123874445241170, 5.08147379778681274403972326026, 5.36999767415097759849494969725, 5.74324186231419842167196673709, 6.22141577204021419068447328967, 6.93762712238470241735905450364, 7.06822993089736014769140693933, 7.30230905447808388992626924352, 8.145413864702020628307011276755, 8.221926532255516463626822504672, 8.378993031342201575205071941172, 9.077800964433028957343356334296, 9.141931480489875094216659074786, 9.806615495525444337359550466908, 10.33476274392271626657055836712, 10.74467976594632696587611396444, 10.75797221882905724367195367605, 10.79385868558341383901081913603, 11.35561803397674411099398316399
