| L(s) = 1 | + 3.66e11·2-s + 9.66e22·4-s − 6.96e25·5-s − 9.19e30·7-s + 2.15e34·8-s − 2.55e37·10-s + 2.04e39·11-s + 1.82e41·13-s − 3.36e42·14-s + 4.25e45·16-s − 1.87e46·17-s + 3.33e47·19-s − 6.72e48·20-s + 7.50e50·22-s − 3.34e50·23-s − 2.16e52·25-s + 6.68e52·26-s − 8.88e53·28-s + 4.55e54·29-s + 7.56e55·31-s + 7.46e56·32-s − 6.88e57·34-s + 6.39e56·35-s + 7.48e58·37-s + 1.22e59·38-s − 1.50e60·40-s + 3.93e60·41-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 2.55·4-s − 0.427·5-s − 0.187·7-s + 2.93·8-s − 0.807·10-s + 1.81·11-s + 0.307·13-s − 0.352·14-s + 2.98·16-s − 1.35·17-s + 0.371·19-s − 1.09·20-s + 3.42·22-s − 0.288·23-s − 0.816·25-s + 0.580·26-s − 0.478·28-s + 0.658·29-s + 0.896·31-s + 2.68·32-s − 2.55·34-s + 0.0800·35-s + 1.16·37-s + 0.700·38-s − 1.25·40-s + 1.30·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(76-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+75/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(38)\) |
\(\approx\) |
\(10.14022709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.14022709\) |
| \(L(\frac{77}{2})\) |
|
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 | \( 1 - 3.66e11T + 3.77e22T^{2} \) |
| 5 | \( 1 + 6.96e25T + 2.64e52T^{2} \) |
| 7 | \( 1 + 9.19e30T + 2.41e63T^{2} \) |
| 11 | \( 1 - 2.04e39T + 1.27e78T^{2} \) |
| 13 | \( 1 - 1.82e41T + 3.51e83T^{2} \) |
| 17 | \( 1 + 1.87e46T + 1.92e92T^{2} \) |
| 19 | \( 1 - 3.33e47T + 8.06e95T^{2} \) |
| 23 | \( 1 + 3.34e50T + 1.34e102T^{2} \) |
| 29 | \( 1 - 4.55e54T + 4.78e109T^{2} \) |
| 31 | \( 1 - 7.56e55T + 7.11e111T^{2} \) |
| 37 | \( 1 - 7.48e58T + 4.12e117T^{2} \) |
| 41 | \( 1 - 3.93e60T + 9.09e120T^{2} \) |
| 43 | \( 1 - 2.53e61T + 3.23e122T^{2} \) |
| 47 | \( 1 - 6.14e62T + 2.55e125T^{2} \) |
| 53 | \( 1 + 6.80e64T + 2.09e129T^{2} \) |
| 59 | \( 1 - 1.05e66T + 6.51e132T^{2} \) |
| 61 | \( 1 + 1.56e67T + 7.93e133T^{2} \) |
| 67 | \( 1 + 2.58e68T + 9.02e136T^{2} \) |
| 71 | \( 1 + 2.66e68T + 6.98e138T^{2} \) |
| 73 | \( 1 - 7.27e69T + 5.61e139T^{2} \) |
| 79 | \( 1 - 2.33e71T + 2.09e142T^{2} \) |
| 83 | \( 1 + 9.21e71T + 8.52e143T^{2} \) |
| 89 | \( 1 - 1.79e72T + 1.60e146T^{2} \) |
| 97 | \( 1 - 4.61e74T + 1.01e149T^{2} \) |
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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
−10.96739949507782211362035872210, −9.272507351851652473330364928355, −7.66341466500232401560332189434, −6.49310280026516194175662576056, −6.05470167879215918121756922314, −4.46603005789389377733386045132, −4.12360900194594392370377923832, −3.09062147712314889886825272892, −2.03656030755018789977449097512, −0.934458180890849293064929241155,
0.934458180890849293064929241155, 2.03656030755018789977449097512, 3.09062147712314889886825272892, 4.12360900194594392370377923832, 4.46603005789389377733386045132, 6.05470167879215918121756922314, 6.49310280026516194175662576056, 7.66341466500232401560332189434, 9.272507351851652473330364928355, 10.96739949507782211362035872210
