| L(s) = 1 | − 3.19e10·2-s − 1.66e16·3-s + 4.33e20·4-s − 1.19e23·5-s + 5.33e26·6-s + 2.02e29·7-s + 5.01e30·8-s + 2.78e32·9-s + 3.83e33·10-s − 9.10e35·11-s − 7.23e36·12-s − 3.11e37·13-s − 6.48e39·14-s + 1.99e39·15-s − 4.16e41·16-s + 2.29e42·17-s − 8.89e42·18-s + 4.71e43·19-s − 5.19e43·20-s − 3.37e45·21-s + 2.91e46·22-s − 1.39e47·23-s − 8.35e46·24-s − 1.67e48·25-s + 9.95e47·26-s − 4.63e48·27-s + 8.78e49·28-s + ⋯ |
| L(s) = 1 | − 1.31·2-s − 0.577·3-s + 0.734·4-s − 0.0921·5-s + 0.760·6-s + 1.41·7-s + 0.349·8-s + 0.333·9-s + 0.121·10-s − 1.07·11-s − 0.424·12-s − 0.115·13-s − 1.86·14-s + 0.0531·15-s − 1.19·16-s + 0.813·17-s − 0.439·18-s + 0.359·19-s − 0.0676·20-s − 0.816·21-s + 1.41·22-s − 1.46·23-s − 0.201·24-s − 0.991·25-s + 0.151·26-s − 0.192·27-s + 1.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(70-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+69/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(35)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{71}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.66e16T \) |
| good | 2 | \( 1 + 3.19e10T + 5.90e20T^{2} \) |
| 5 | \( 1 + 1.19e23T + 1.69e48T^{2} \) |
| 7 | \( 1 - 2.02e29T + 2.05e58T^{2} \) |
| 11 | \( 1 + 9.10e35T + 7.17e71T^{2} \) |
| 13 | \( 1 + 3.11e37T + 7.27e76T^{2} \) |
| 17 | \( 1 - 2.29e42T + 7.96e84T^{2} \) |
| 19 | \( 1 - 4.71e43T + 1.71e88T^{2} \) |
| 23 | \( 1 + 1.39e47T + 9.10e93T^{2} \) |
| 29 | \( 1 + 1.66e50T + 8.04e100T^{2} \) |
| 31 | \( 1 - 2.79e51T + 8.01e102T^{2} \) |
| 37 | \( 1 - 1.41e54T + 1.60e108T^{2} \) |
| 41 | \( 1 + 2.72e55T + 1.91e111T^{2} \) |
| 43 | \( 1 - 1.02e56T + 5.12e112T^{2} \) |
| 47 | \( 1 - 5.64e57T + 2.37e115T^{2} \) |
| 53 | \( 1 - 3.40e59T + 9.44e118T^{2} \) |
| 59 | \( 1 + 2.01e61T + 1.54e122T^{2} \) |
| 61 | \( 1 + 4.23e61T + 1.54e123T^{2} \) |
| 67 | \( 1 - 7.65e62T + 9.98e125T^{2} \) |
| 71 | \( 1 - 4.39e63T + 5.45e127T^{2} \) |
| 73 | \( 1 - 2.02e64T + 3.70e128T^{2} \) |
| 79 | \( 1 + 5.59e65T + 8.63e130T^{2} \) |
| 83 | \( 1 - 1.81e63T + 2.60e132T^{2} \) |
| 89 | \( 1 - 1.30e67T + 3.22e134T^{2} \) |
| 97 | \( 1 + 5.99e68T + 1.22e137T^{2} \) |
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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
−11.90791708690749701967787057611, −10.80565342394047976035902318312, −9.803037960182357165094800305061, −8.122728866664748208894737028646, −7.59719669879101282167371950748, −5.62290442526079231187906511866, −4.40065807158225104821366751367, −2.16545218842423932866201896474, −1.10889030032781989446526391078, 0,
1.10889030032781989446526391078, 2.16545218842423932866201896474, 4.40065807158225104821366751367, 5.62290442526079231187906511866, 7.59719669879101282167371950748, 8.122728866664748208894737028646, 9.803037960182357165094800305061, 10.80565342394047976035902318312, 11.90791708690749701967787057611
