| L(s) = 1 | + 3.35e9·2-s + 5.55e15·3-s − 1.36e20·4-s − 3.80e23·5-s + 1.86e25·6-s + 2.28e27·7-s − 9.51e29·8-s + 3.09e31·9-s − 1.27e33·10-s + 6.32e34·11-s − 7.57e35·12-s + 1.06e37·13-s + 7.65e36·14-s − 2.11e39·15-s + 1.69e40·16-s + 1.88e41·17-s + 1.03e41·18-s + 7.47e42·19-s + 5.18e43·20-s + 1.26e43·21-s + 2.12e44·22-s − 9.84e44·23-s − 5.29e45·24-s + 7.71e46·25-s + 3.58e46·26-s + 1.71e47·27-s − 3.11e47·28-s + ⋯ |
| L(s) = 1 | + 0.275·2-s + 0.577·3-s − 0.923·4-s − 1.46·5-s + 0.159·6-s + 0.111·7-s − 0.530·8-s + 0.333·9-s − 0.403·10-s + 0.821·11-s − 0.533·12-s + 0.514·13-s + 0.0308·14-s − 0.844·15-s + 0.777·16-s + 1.13·17-s + 0.0919·18-s + 1.08·19-s + 1.35·20-s + 0.0644·21-s + 0.226·22-s − 0.237·23-s − 0.306·24-s + 1.13·25-s + 0.142·26-s + 0.192·27-s − 0.103·28-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
| \(L(34)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{69}{2})\) |
|
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 | \( 1 - 3.35e9T + 1.47e20T^{2} \) |
| 5 | \( 1 + 3.80e23T + 6.77e46T^{2} \) |
| 7 | \( 1 - 2.28e27T + 4.18e56T^{2} \) |
| 11 | \( 1 - 6.32e34T + 5.93e69T^{2} \) |
| 13 | \( 1 - 1.06e37T + 4.30e74T^{2} \) |
| 17 | \( 1 - 1.88e41T + 2.75e82T^{2} \) |
| 19 | \( 1 - 7.47e42T + 4.74e85T^{2} \) |
| 23 | \( 1 + 9.84e44T + 1.72e91T^{2} \) |
| 29 | \( 1 + 1.61e49T + 9.56e97T^{2} \) |
| 31 | \( 1 + 1.42e50T + 8.34e99T^{2} \) |
| 37 | \( 1 + 3.12e52T + 1.17e105T^{2} \) |
| 41 | \( 1 - 3.16e53T + 1.13e108T^{2} \) |
| 43 | \( 1 + 4.14e54T + 2.76e109T^{2} \) |
| 47 | \( 1 - 7.48e54T + 1.07e112T^{2} \) |
| 53 | \( 1 - 2.48e57T + 3.36e115T^{2} \) |
| 59 | \( 1 - 2.25e59T + 4.43e118T^{2} \) |
| 61 | \( 1 + 1.12e59T + 4.14e119T^{2} \) |
| 67 | \( 1 - 2.53e61T + 2.22e122T^{2} \) |
| 71 | \( 1 + 1.31e62T + 1.08e124T^{2} \) |
| 73 | \( 1 - 4.80e62T + 6.96e124T^{2} \) |
| 79 | \( 1 + 4.20e63T + 1.38e127T^{2} \) |
| 83 | \( 1 - 2.02e64T + 3.78e128T^{2} \) |
| 89 | \( 1 + 2.92e65T + 4.06e130T^{2} \) |
| 97 | \( 1 + 7.09e66T + 1.29e133T^{2} \) |
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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
−12.63307717126230572218126679139, −11.47516429718571760113595800126, −9.541625557295795310888053353236, −8.389604715302618283516240734331, −7.37309994129585732330357929322, −5.33736438097722090808676173000, −3.80699273650298172351026620070, −3.52619851462928586190386531393, −1.26244158308576328047536314886, 0,
1.26244158308576328047536314886, 3.52619851462928586190386531393, 3.80699273650298172351026620070, 5.33736438097722090808676173000, 7.37309994129585732330357929322, 8.389604715302618283516240734331, 9.541625557295795310888053353236, 11.47516429718571760113595800126, 12.63307717126230572218126679139
