| L(s) = 1 | + 6.58e13·2-s + 3.71e20·3-s − 1.54e29·4-s − 8.33e33·5-s + 2.44e34·6-s + 1.07e41·7-s − 2.05e43·8-s − 1.90e46·9-s − 5.48e47·10-s + 3.04e50·11-s − 5.72e49·12-s + 1.64e54·13-s + 7.05e54·14-s − 3.09e54·15-s + 2.30e58·16-s + 4.63e59·17-s − 1.25e60·18-s − 1.62e62·19-s + 1.28e63·20-s + 3.98e61·21-s + 2.00e64·22-s − 4.46e65·23-s − 7.64e63·24-s + 6.30e66·25-s + 1.08e68·26-s − 1.41e67·27-s − 1.65e70·28-s + ⋯ |
| L(s) = 1 | + 0.165·2-s + 0.00268·3-s − 0.972·4-s − 1.04·5-s + 0.000444·6-s + 1.10·7-s − 0.326·8-s − 0.999·9-s − 0.173·10-s + 0.944·11-s − 0.00261·12-s + 1.54·13-s + 0.182·14-s − 0.00281·15-s + 0.918·16-s + 0.974·17-s − 0.165·18-s − 1.55·19-s + 1.02·20-s + 0.00296·21-s + 0.156·22-s − 0.403·23-s − 0.000876·24-s + 0.0999·25-s + 0.255·26-s − 0.00537·27-s − 1.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(98-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+97/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(49)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{99}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 6.58e13T + 1.58e29T^{2} \) |
| 3 | \( 1 - 3.71e20T + 1.90e46T^{2} \) |
| 5 | \( 1 + 8.33e33T + 6.31e67T^{2} \) |
| 7 | \( 1 - 1.07e41T + 9.42e81T^{2} \) |
| 11 | \( 1 - 3.04e50T + 1.03e101T^{2} \) |
| 13 | \( 1 - 1.64e54T + 1.12e108T^{2} \) |
| 17 | \( 1 - 4.63e59T + 2.25e119T^{2} \) |
| 19 | \( 1 + 1.62e62T + 1.09e124T^{2} \) |
| 23 | \( 1 + 4.46e65T + 1.22e132T^{2} \) |
| 29 | \( 1 - 9.12e70T + 7.12e141T^{2} \) |
| 31 | \( 1 + 2.66e72T + 4.59e144T^{2} \) |
| 37 | \( 1 - 1.66e75T + 1.30e152T^{2} \) |
| 41 | \( 1 + 4.42e77T + 2.75e156T^{2} \) |
| 43 | \( 1 - 2.04e78T + 2.79e158T^{2} \) |
| 47 | \( 1 + 4.34e80T + 1.56e162T^{2} \) |
| 53 | \( 1 - 6.04e83T + 1.79e167T^{2} \) |
| 59 | \( 1 + 1.47e86T + 5.92e171T^{2} \) |
| 61 | \( 1 + 3.98e86T + 1.50e173T^{2} \) |
| 67 | \( 1 + 3.81e88T + 1.34e177T^{2} \) |
| 71 | \( 1 + 7.68e88T + 3.73e179T^{2} \) |
| 73 | \( 1 - 2.34e90T + 5.52e180T^{2} \) |
| 79 | \( 1 - 5.91e91T + 1.17e184T^{2} \) |
| 83 | \( 1 + 2.21e93T + 1.41e186T^{2} \) |
| 89 | \( 1 + 3.18e94T + 1.23e189T^{2} \) |
| 97 | \( 1 + 1.10e96T + 5.21e192T^{2} \) |
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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
−13.96516983069555111388914297680, −12.12741437767863815501854333481, −10.98360792004159420589764865755, −8.752478729443882923935677937935, −8.102678262834135042441739126018, −5.91740434306754388406661234748, −4.39766914290604370761457215643, −3.52328749558291619005464272133, −1.30860019250332454201757636307, 0,
1.30860019250332454201757636307, 3.52328749558291619005464272133, 4.39766914290604370761457215643, 5.91740434306754388406661234748, 8.102678262834135042441739126018, 8.752478729443882923935677937935, 10.98360792004159420589764865755, 12.12741437767863815501854333481, 13.96516983069555111388914297680
