| L(s) = 1 | + 4.67e16·2-s + 3.08e25·3-s + 1.53e33·4-s − 1.66e38·5-s + 1.44e42·6-s + 8.52e45·7-s + 4.16e49·8-s − 9.19e51·9-s − 7.76e54·10-s − 4.17e56·11-s + 4.74e58·12-s − 8.29e60·13-s + 3.98e62·14-s − 5.12e63·15-s + 9.47e65·16-s − 6.45e65·17-s − 4.30e68·18-s − 6.23e69·19-s − 2.55e71·20-s + 2.62e71·21-s − 1.95e73·22-s + 2.48e74·23-s + 1.28e75·24-s + 1.21e76·25-s − 3.88e77·26-s − 5.96e77·27-s + 1.31e79·28-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 0.306·3-s + 2.37·4-s − 1.33·5-s + 0.562·6-s + 0.746·7-s + 2.51·8-s − 0.906·9-s − 2.45·10-s − 0.732·11-s + 0.725·12-s − 1.61·13-s + 1.37·14-s − 0.409·15-s + 2.25·16-s − 0.0563·17-s − 1.66·18-s − 1.26·19-s − 3.17·20-s + 0.228·21-s − 1.34·22-s + 1.51·23-s + 0.770·24-s + 0.790·25-s − 2.97·26-s − 0.583·27-s + 1.76·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(110-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+54.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(55)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{111}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 4.67e16T + 6.49e32T^{2} \) |
| 3 | \( 1 - 3.08e25T + 1.01e52T^{2} \) |
| 5 | \( 1 + 1.66e38T + 1.54e76T^{2} \) |
| 7 | \( 1 - 8.52e45T + 1.30e92T^{2} \) |
| 11 | \( 1 + 4.17e56T + 3.24e113T^{2} \) |
| 13 | \( 1 + 8.29e60T + 2.62e121T^{2} \) |
| 17 | \( 1 + 6.45e65T + 1.31e134T^{2} \) |
| 19 | \( 1 + 6.23e69T + 2.42e139T^{2} \) |
| 23 | \( 1 - 2.48e74T + 2.68e148T^{2} \) |
| 29 | \( 1 - 1.15e79T + 2.51e159T^{2} \) |
| 31 | \( 1 + 2.19e81T + 3.61e162T^{2} \) |
| 37 | \( 1 - 5.13e85T + 8.58e170T^{2} \) |
| 41 | \( 1 + 1.09e88T + 6.21e175T^{2} \) |
| 43 | \( 1 + 1.31e89T + 1.11e178T^{2} \) |
| 47 | \( 1 - 4.22e89T + 1.81e182T^{2} \) |
| 53 | \( 1 - 5.60e93T + 8.83e187T^{2} \) |
| 59 | \( 1 - 2.61e96T + 1.05e193T^{2} \) |
| 61 | \( 1 - 2.27e96T + 3.98e194T^{2} \) |
| 67 | \( 1 + 1.52e98T + 1.10e199T^{2} \) |
| 71 | \( 1 + 2.16e100T + 6.12e201T^{2} \) |
| 73 | \( 1 + 1.09e101T + 1.26e203T^{2} \) |
| 79 | \( 1 + 1.07e103T + 6.93e206T^{2} \) |
| 83 | \( 1 - 3.89e104T + 1.51e209T^{2} \) |
| 89 | \( 1 + 1.88e105T + 3.04e212T^{2} \) |
| 97 | \( 1 + 3.05e108T + 3.61e216T^{2} \) |
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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
−13.04475624788368262617403038155, −11.86168082349017383599296530333, −11.00155185100921386454067308314, −8.144908194986034085858207670834, −7.06432073820129164691666467664, −5.25191499314382153134292664489, −4.45768567347611492414493694150, −3.18020390465006111019677626795, −2.24762534544577145625121646197, 0,
2.24762534544577145625121646197, 3.18020390465006111019677626795, 4.45768567347611492414493694150, 5.25191499314382153134292664489, 7.06432073820129164691666467664, 8.144908194986034085858207670834, 11.00155185100921386454067308314, 11.86168082349017383599296530333, 13.04475624788368262617403038155
