| L(s) = 1 | + 1.15e5·2-s − 1.23e11·4-s − 1.39e13·5-s + 5.04e15·7-s − 3.03e16·8-s − 1.61e18·10-s + 1.87e19·11-s − 5.91e20·13-s + 5.85e20·14-s + 1.35e22·16-s + 1.12e23·17-s + 4.38e23·19-s + 1.72e24·20-s + 2.17e24·22-s − 2.25e25·23-s + 1.20e26·25-s − 6.86e25·26-s − 6.25e26·28-s + 7.84e26·29-s − 2.88e27·31-s + 5.73e27·32-s + 1.30e28·34-s − 7.01e28·35-s − 1.72e28·37-s + 5.09e28·38-s + 4.21e29·40-s + 4.53e29·41-s + ⋯ |
| L(s) = 1 | + 0.312·2-s − 0.902·4-s − 1.63·5-s + 1.17·7-s − 0.595·8-s − 0.509·10-s + 1.01·11-s − 1.46·13-s + 0.366·14-s + 0.716·16-s + 1.94·17-s + 0.967·19-s + 1.47·20-s + 0.317·22-s − 1.44·23-s + 1.65·25-s − 0.456·26-s − 1.05·28-s + 0.691·29-s − 0.741·31-s + 0.819·32-s + 0.608·34-s − 1.91·35-s − 0.167·37-s + 0.302·38-s + 0.969·40-s + 0.660·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(19)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{39}{2})\) |
|
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 | \( 1 - 1.15e5T + 1.37e11T^{2} \) |
| 5 | \( 1 + 1.39e13T + 7.27e25T^{2} \) |
| 7 | \( 1 - 5.04e15T + 1.85e31T^{2} \) |
| 11 | \( 1 - 1.87e19T + 3.40e38T^{2} \) |
| 13 | \( 1 + 5.91e20T + 1.64e41T^{2} \) |
| 17 | \( 1 - 1.12e23T + 3.36e45T^{2} \) |
| 19 | \( 1 - 4.38e23T + 2.06e47T^{2} \) |
| 23 | \( 1 + 2.25e25T + 2.42e50T^{2} \) |
| 29 | \( 1 - 7.84e26T + 1.28e54T^{2} \) |
| 31 | \( 1 + 2.88e27T + 1.51e55T^{2} \) |
| 37 | \( 1 + 1.72e28T + 1.05e58T^{2} \) |
| 41 | \( 1 - 4.53e29T + 4.70e59T^{2} \) |
| 43 | \( 1 + 1.58e30T + 2.74e60T^{2} \) |
| 47 | \( 1 + 1.10e31T + 7.37e61T^{2} \) |
| 53 | \( 1 - 4.79e31T + 6.28e63T^{2} \) |
| 59 | \( 1 - 1.00e32T + 3.32e65T^{2} \) |
| 61 | \( 1 - 3.87e31T + 1.14e66T^{2} \) |
| 67 | \( 1 - 2.81e33T + 3.67e67T^{2} \) |
| 71 | \( 1 + 8.65e33T + 3.13e68T^{2} \) |
| 73 | \( 1 + 4.05e34T + 8.76e68T^{2} \) |
| 79 | \( 1 + 3.26e34T + 1.63e70T^{2} \) |
| 83 | \( 1 - 2.87e34T + 1.01e71T^{2} \) |
| 89 | \( 1 - 9.54e35T + 1.34e72T^{2} \) |
| 97 | \( 1 + 4.38e36T + 3.24e73T^{2} \) |
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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
−12.14164053444100598244425634265, −11.81345609231217408269247476069, −9.814985350576631826817840406166, −8.236040378576736480271964031000, −7.51480773959274282313268616507, −5.26807729805397711654190175119, −4.30593392616166243675503422801, −3.35693723319661237016483807383, −1.19278335569823834348412650732, 0,
1.19278335569823834348412650732, 3.35693723319661237016483807383, 4.30593392616166243675503422801, 5.26807729805397711654190175119, 7.51480773959274282313268616507, 8.236040378576736480271964031000, 9.814985350576631826817840406166, 11.81345609231217408269247476069, 12.14164053444100598244425634265
