| L(s) = 1 | − 3.67e11·2-s + 9.72e22·4-s − 2.43e26·5-s + 1.21e31·7-s − 2.18e34·8-s + 8.95e37·10-s + 5.65e38·11-s + 5.30e40·13-s − 4.46e42·14-s + 4.35e45·16-s − 1.14e45·17-s + 1.24e48·19-s − 2.36e49·20-s − 2.07e50·22-s − 8.02e50·23-s + 3.28e52·25-s − 1.94e52·26-s + 1.18e54·28-s − 8.53e54·29-s − 6.18e55·31-s − 7.74e56·32-s + 4.21e56·34-s − 2.96e57·35-s − 5.98e58·37-s − 4.58e59·38-s + 5.32e60·40-s + 5.32e60·41-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 2.57·4-s − 1.49·5-s + 0.247·7-s − 2.97·8-s + 2.83·10-s + 0.501·11-s + 0.0894·13-s − 0.467·14-s + 3.05·16-s − 0.0828·17-s + 1.38·19-s − 3.85·20-s − 0.947·22-s − 0.691·23-s + 1.24·25-s − 0.169·26-s + 0.636·28-s − 1.23·29-s − 0.733·31-s − 2.79·32-s + 0.156·34-s − 0.370·35-s − 0.932·37-s − 2.62·38-s + 4.45·40-s + 1.76·41-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
| \(L(38)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{77}{2})\) |
|
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 | \( 1 + 3.67e11T + 3.77e22T^{2} \) |
| 5 | \( 1 + 2.43e26T + 2.64e52T^{2} \) |
| 7 | \( 1 - 1.21e31T + 2.41e63T^{2} \) |
| 11 | \( 1 - 5.65e38T + 1.27e78T^{2} \) |
| 13 | \( 1 - 5.30e40T + 3.51e83T^{2} \) |
| 17 | \( 1 + 1.14e45T + 1.92e92T^{2} \) |
| 19 | \( 1 - 1.24e48T + 8.06e95T^{2} \) |
| 23 | \( 1 + 8.02e50T + 1.34e102T^{2} \) |
| 29 | \( 1 + 8.53e54T + 4.78e109T^{2} \) |
| 31 | \( 1 + 6.18e55T + 7.11e111T^{2} \) |
| 37 | \( 1 + 5.98e58T + 4.12e117T^{2} \) |
| 41 | \( 1 - 5.32e60T + 9.09e120T^{2} \) |
| 43 | \( 1 - 2.59e61T + 3.23e122T^{2} \) |
| 47 | \( 1 - 5.76e62T + 2.55e125T^{2} \) |
| 53 | \( 1 + 5.92e63T + 2.09e129T^{2} \) |
| 59 | \( 1 + 2.82e66T + 6.51e132T^{2} \) |
| 61 | \( 1 - 5.43e66T + 7.93e133T^{2} \) |
| 67 | \( 1 + 2.61e68T + 9.02e136T^{2} \) |
| 71 | \( 1 + 1.78e67T + 6.98e138T^{2} \) |
| 73 | \( 1 + 1.09e70T + 5.61e139T^{2} \) |
| 79 | \( 1 + 1.49e71T + 2.09e142T^{2} \) |
| 83 | \( 1 + 3.39e71T + 8.52e143T^{2} \) |
| 89 | \( 1 + 2.45e72T + 1.60e146T^{2} \) |
| 97 | \( 1 - 1.36e74T + 1.01e149T^{2} \) |
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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
−9.505884550787173223410860837602, −8.697220938162827584130475707133, −7.55603045679181518858312178934, −7.36775755464504859668786662297, −5.85513004420522033033001403256, −4.04771227342271623544713294680, −3.01197969896575095959713994522, −1.71620573632080962516649309151, −0.801277384034727705860552902398, 0,
0.801277384034727705860552902398, 1.71620573632080962516649309151, 3.01197969896575095959713994522, 4.04771227342271623544713294680, 5.85513004420522033033001403256, 7.36775755464504859668786662297, 7.55603045679181518858312178934, 8.697220938162827584130475707133, 9.505884550787173223410860837602
