| L(s) = 1 | + 9.03e9·2-s − 3.76e22·4-s − 2.32e25·5-s + 5.93e31·7-s − 6.81e32·8-s − 2.09e35·10-s + 1.90e39·11-s + 6.51e41·13-s + 5.35e41·14-s + 1.41e45·16-s − 1.17e46·17-s + 6.71e45·19-s + 8.75e47·20-s + 1.71e49·22-s − 1.17e51·23-s − 2.59e52·25-s + 5.88e51·26-s − 2.23e54·28-s − 4.12e53·29-s − 1.59e56·31-s + 3.85e55·32-s − 1.05e56·34-s − 1.37e57·35-s − 3.73e57·37-s + 6.06e55·38-s + 1.58e58·40-s + 4.94e60·41-s + ⋯ |
| L(s) = 1 | + 0.0464·2-s − 0.997·4-s − 0.142·5-s + 1.20·7-s − 0.0928·8-s − 0.00663·10-s + 1.68·11-s + 1.09·13-s + 0.0561·14-s + 0.993·16-s − 0.844·17-s + 0.00747·19-s + 0.142·20-s + 0.0783·22-s − 1.01·23-s − 0.979·25-s + 0.0511·26-s − 1.20·28-s − 0.0596·29-s − 1.89·31-s + 0.139·32-s − 0.0392·34-s − 0.172·35-s − 0.0582·37-s + 0.000347·38-s + 0.0132·40-s + 1.64·41-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 | \( 1 - 9.03e9T + 3.77e22T^{2} \) |
| 5 | \( 1 + 2.32e25T + 2.64e52T^{2} \) |
| 7 | \( 1 - 5.93e31T + 2.41e63T^{2} \) |
| 11 | \( 1 - 1.90e39T + 1.27e78T^{2} \) |
| 13 | \( 1 - 6.51e41T + 3.51e83T^{2} \) |
| 17 | \( 1 + 1.17e46T + 1.92e92T^{2} \) |
| 19 | \( 1 - 6.71e45T + 8.06e95T^{2} \) |
| 23 | \( 1 + 1.17e51T + 1.34e102T^{2} \) |
| 29 | \( 1 + 4.12e53T + 4.78e109T^{2} \) |
| 31 | \( 1 + 1.59e56T + 7.11e111T^{2} \) |
| 37 | \( 1 + 3.73e57T + 4.12e117T^{2} \) |
| 41 | \( 1 - 4.94e60T + 9.09e120T^{2} \) |
| 43 | \( 1 + 1.37e61T + 3.23e122T^{2} \) |
| 47 | \( 1 + 4.17e62T + 2.55e125T^{2} \) |
| 53 | \( 1 + 4.77e64T + 2.09e129T^{2} \) |
| 59 | \( 1 - 3.22e66T + 6.51e132T^{2} \) |
| 61 | \( 1 - 1.28e67T + 7.93e133T^{2} \) |
| 67 | \( 1 - 8.16e67T + 9.02e136T^{2} \) |
| 71 | \( 1 + 3.34e69T + 6.98e138T^{2} \) |
| 73 | \( 1 - 2.71e69T + 5.61e139T^{2} \) |
| 79 | \( 1 - 8.63e70T + 2.09e142T^{2} \) |
| 83 | \( 1 - 3.24e71T + 8.52e143T^{2} \) |
| 89 | \( 1 - 1.12e73T + 1.60e146T^{2} \) |
| 97 | \( 1 + 5.71e72T + 1.01e149T^{2} \) |
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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
−9.500173592452332807953210345933, −8.710489884552642580548797980779, −7.86229838806648283992197482958, −6.39782631109671681445064619293, −5.32944970265294582861548201676, −4.09951559020741662440782403103, −3.81201045414142830750927497037, −1.86555728852281599180708342470, −1.17848687845905382779707334215, 0,
1.17848687845905382779707334215, 1.86555728852281599180708342470, 3.81201045414142830750927497037, 4.09951559020741662440782403103, 5.32944970265294582861548201676, 6.39782631109671681445064619293, 7.86229838806648283992197482958, 8.710489884552642580548797980779, 9.500173592452332807953210345933
