| L(s) = 1 | − 1.17e10·2-s − 8.92e18·4-s − 8.66e22·5-s + 3.38e28·7-s + 1.84e30·8-s + 1.02e33·10-s + 2.84e34·11-s + 5.77e36·13-s − 3.99e38·14-s − 2.03e40·16-s + 1.58e41·17-s + 1.30e43·19-s + 7.73e41·20-s − 3.35e44·22-s − 5.95e44·23-s − 6.02e46·25-s − 6.80e46·26-s − 3.02e47·28-s + 1.88e48·29-s − 9.76e49·31-s − 3.19e49·32-s − 1.87e51·34-s − 2.93e51·35-s − 2.00e52·37-s − 1.53e53·38-s − 1.59e53·40-s − 1.76e54·41-s + ⋯ |
| L(s) = 1 | − 0.969·2-s − 0.0605·4-s − 0.332·5-s + 1.65·7-s + 1.02·8-s + 0.322·10-s + 0.369·11-s + 0.278·13-s − 1.60·14-s − 0.935·16-s + 0.957·17-s + 1.89·19-s + 0.0201·20-s − 0.358·22-s − 0.143·23-s − 0.889·25-s − 0.269·26-s − 0.100·28-s + 0.192·29-s − 1.06·31-s − 0.120·32-s − 0.928·34-s − 0.551·35-s − 0.586·37-s − 1.83·38-s − 0.342·40-s − 1.65·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(68-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+67/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(34)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{69}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 1.17e10T + 1.47e20T^{2} \) |
| 5 | \( 1 + 8.66e22T + 6.77e46T^{2} \) |
| 7 | \( 1 - 3.38e28T + 4.18e56T^{2} \) |
| 11 | \( 1 - 2.84e34T + 5.93e69T^{2} \) |
| 13 | \( 1 - 5.77e36T + 4.30e74T^{2} \) |
| 17 | \( 1 - 1.58e41T + 2.75e82T^{2} \) |
| 19 | \( 1 - 1.30e43T + 4.74e85T^{2} \) |
| 23 | \( 1 + 5.95e44T + 1.72e91T^{2} \) |
| 29 | \( 1 - 1.88e48T + 9.56e97T^{2} \) |
| 31 | \( 1 + 9.76e49T + 8.34e99T^{2} \) |
| 37 | \( 1 + 2.00e52T + 1.17e105T^{2} \) |
| 41 | \( 1 + 1.76e54T + 1.13e108T^{2} \) |
| 43 | \( 1 - 5.65e54T + 2.76e109T^{2} \) |
| 47 | \( 1 + 1.13e56T + 1.07e112T^{2} \) |
| 53 | \( 1 + 5.57e57T + 3.36e115T^{2} \) |
| 59 | \( 1 + 2.54e59T + 4.43e118T^{2} \) |
| 61 | \( 1 + 4.65e59T + 4.14e119T^{2} \) |
| 67 | \( 1 - 7.13e59T + 2.22e122T^{2} \) |
| 71 | \( 1 + 1.41e62T + 1.08e124T^{2} \) |
| 73 | \( 1 + 2.86e62T + 6.96e124T^{2} \) |
| 79 | \( 1 - 2.81e63T + 1.38e127T^{2} \) |
| 83 | \( 1 + 3.57e64T + 3.78e128T^{2} \) |
| 89 | \( 1 + 2.86e64T + 4.06e130T^{2} \) |
| 97 | \( 1 - 4.69e66T + 1.29e133T^{2} \) |
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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
−10.00221858072132091086593298831, −8.928459247613125926636802177867, −7.914386237351757455610392479992, −7.42527701491487736015126937457, −5.50036010460054118727197067399, −4.61046089882226900514317116674, −3.43134000680604443362282777893, −1.61765155533672895915744309996, −1.25900849245042793608010712720, 0,
1.25900849245042793608010712720, 1.61765155533672895915744309996, 3.43134000680604443362282777893, 4.61046089882226900514317116674, 5.50036010460054118727197067399, 7.42527701491487736015126937457, 7.914386237351757455610392479992, 8.928459247613125926636802177867, 10.00221858072132091086593298831
