L(s) = 1 | + 1.17e10·2-s − 5.55e15·3-s − 8.92e18·4-s + 8.66e22·5-s − 6.54e25·6-s + 3.38e28·7-s − 1.84e30·8-s + 3.09e31·9-s + 1.02e33·10-s − 2.84e34·11-s + 4.96e34·12-s + 5.77e36·13-s + 3.99e38·14-s − 4.81e38·15-s − 2.03e40·16-s − 1.58e41·17-s + 3.63e41·18-s + 1.30e43·19-s − 7.73e41·20-s − 1.88e44·21-s − 3.35e44·22-s + 5.95e44·23-s + 1.02e46·24-s − 6.02e46·25-s + 6.80e46·26-s − 1.71e47·27-s − 3.02e47·28-s + ⋯ |
L(s) = 1 | + 0.969·2-s − 0.577·3-s − 0.0605·4-s + 0.332·5-s − 0.559·6-s + 1.65·7-s − 1.02·8-s + 0.333·9-s + 0.322·10-s − 0.369·11-s + 0.0349·12-s + 0.278·13-s + 1.60·14-s − 0.192·15-s − 0.935·16-s − 0.957·17-s + 0.323·18-s + 1.89·19-s − 0.0201·20-s − 0.956·21-s − 0.358·22-s + 0.143·23-s + 0.593·24-s − 0.889·25-s + 0.269·26-s − 0.192·27-s − 0.100·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(68-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+67/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(34)\) |
\(\approx\) |
\(3.254019572\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.254019572\) |
\(L(\frac{69}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 5.55e15T \) |
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
−13.42986059716772123283000144034, −11.95519150438361005747514377227, −10.97544649159166527124277713177, −9.139860877135346242982796522328, −7.54450914171609948342417051246, −5.72967124991006947899200807819, −5.05585809684835227487684422942, −3.94894473150087184723454794389, −2.21130381419611809885876262625, −0.829137807603112330932751473845,
0.829137807603112330932751473845, 2.21130381419611809885876262625, 3.94894473150087184723454794389, 5.05585809684835227487684422942, 5.72967124991006947899200807819, 7.54450914171609948342417051246, 9.139860877135346242982796522328, 10.97544649159166527124277713177, 11.95519150438361005747514377227, 13.42986059716772123283000144034