| L(s) = 1 | + 18.7·3-s + 1.16e3i·5-s + (2.73e3 − 5.73e3i)7-s − 1.93e4·9-s − 2.83e4i·11-s + 1.18e5i·13-s + 2.19e4i·15-s − 1.31e5i·17-s + 8.87e5·19-s + (5.13e4 − 1.07e5i)21-s − 1.24e5i·23-s + 5.85e5·25-s − 7.33e5·27-s + 7.07e5·29-s − 6.60e6·31-s + ⋯ |
| L(s) = 1 | + 0.133·3-s + 0.836i·5-s + (0.429 − 0.902i)7-s − 0.982·9-s − 0.584i·11-s + 1.14i·13-s + 0.112i·15-s − 0.382i·17-s + 1.56·19-s + (0.0576 − 0.120i)21-s − 0.0928i·23-s + 0.300·25-s − 0.265·27-s + 0.185·29-s − 1.28·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.884086416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.884086416\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.73e3 + 5.73e3i)T \) |
| good | 3 | \( 1 - 18.7T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.16e3iT - 1.95e6T^{2} \) |
| 11 | \( 1 + 2.83e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 1.18e5iT - 1.06e10T^{2} \) |
| 17 | \( 1 + 1.31e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 - 8.87e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.24e5iT - 1.80e12T^{2} \) |
| 29 | \( 1 - 7.07e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.60e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.09e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.80e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 - 2.27e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 - 1.10e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.43e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.27e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.33e5iT - 1.16e16T^{2} \) |
| 67 | \( 1 - 1.77e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 - 3.23e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 - 2.07e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 - 4.45e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 - 3.35e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.63e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + 5.94e8iT - 7.60e17T^{2} \) |
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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
−11.54659128978352902193099082091, −11.25060260742703633089628507521, −9.986798260702055233122133372512, −8.805606883280067534872146796844, −7.56960861003579874330109496640, −6.66216572246511484834914977609, −5.28071453133774233060959517668, −3.76875423826112639203513566889, −2.68538328956372483575653417878, −1.04662241619662367544451024565,
0.54768943175238389360795356360, 2.01721698027747264150075208104, 3.35226510766963281251377419103, 5.13864480919520469502299538608, 5.64048086328681834128728061490, 7.50679791194605461786363566945, 8.553791730719395535580791217401, 9.251094774434756936593668915908, 10.64114930286824485233822689147, 11.89809610223968416453722853191
