| L(s) = 1 | − 3·3-s + 8·4-s + 12i·5-s + 2i·7-s + 9·9-s − 36i·11-s − 24·12-s − 36i·15-s + 64·16-s + 78·17-s − 74i·19-s + 96i·20-s − 6i·21-s + 96·23-s − 19·25-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 4-s + 1.07i·5-s + 0.107i·7-s + 0.333·9-s − 0.986i·11-s − 0.577·12-s − 0.619i·15-s + 16-s + 1.11·17-s − 0.893i·19-s + 1.07i·20-s − 0.0623i·21-s + 0.870·23-s − 0.151·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.214337380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.214337380\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 8T^{2} \) |
| 5 | \( 1 - 12iT - 125T^{2} \) |
| 7 | \( 1 - 2iT - 343T^{2} \) |
| 11 | \( 1 + 36iT - 1.33e3T^{2} \) |
| 17 | \( 1 - 78T + 4.91e3T^{2} \) |
| 19 | \( 1 + 74iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 96T + 1.21e4T^{2} \) |
| 29 | \( 1 - 18T + 2.43e4T^{2} \) |
| 31 | \( 1 - 214iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 286iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 384iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 524T + 7.95e4T^{2} \) |
| 47 | \( 1 - 300iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 558T + 1.48e5T^{2} \) |
| 59 | \( 1 - 576iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 74T + 2.26e5T^{2} \) |
| 67 | \( 1 + 38iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 456iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 682iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 704T + 4.93e5T^{2} \) |
| 83 | \( 1 - 888iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.02e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 110iT - 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.73553956874182388077062748758, −10.10697745563384368033647550658, −8.760890457466282369360590478108, −7.54196263282179002308020102137, −6.85770087983768156585641910833, −6.11274010954071403906120831092, −5.19105321088377414643194425256, −3.42044431367593246000085483793, −2.69597295464047037580463171420, −1.06986029452229188950017688255,
0.922803367563866036483708166567, 1.98206246669603307863181702158, 3.62383725782463126359397899111, 4.92616042861740135160350351223, 5.65954133900115288703694487464, 6.77493939022990281976704007702, 7.58446194570514473587095502389, 8.526892998998024443091679029037, 9.824834403123852412048788783493, 10.31709816777718576249370920241
