| L(s) = 1 | − 4.71e16·2-s − 1.38e26·3-s + 1.57e33·4-s − 1.39e38·5-s + 6.53e42·6-s + 1.59e46·7-s − 4.37e49·8-s + 9.05e51·9-s + 6.57e54·10-s − 5.93e56·11-s − 2.18e59·12-s + 1.89e60·13-s − 7.54e62·14-s + 1.93e64·15-s + 1.04e66·16-s − 1.15e67·17-s − 4.27e68·18-s − 3.88e69·19-s − 2.19e71·20-s − 2.21e72·21-s + 2.79e73·22-s − 1.96e74·23-s + 6.06e75·24-s + 3.99e75·25-s − 8.94e76·26-s + 1.50e77·27-s + 2.52e79·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 1.37·3-s + 2.42·4-s − 1.12·5-s + 2.54·6-s + 1.39·7-s − 2.64·8-s + 0.893·9-s + 2.07·10-s − 1.04·11-s − 3.34·12-s + 0.369·13-s − 2.59·14-s + 1.54·15-s + 2.46·16-s − 1.00·17-s − 1.65·18-s − 0.789·19-s − 2.72·20-s − 1.92·21-s + 1.92·22-s − 1.19·23-s + 3.63·24-s + 0.259·25-s − 0.684·26-s + 0.147·27-s + 3.39·28-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 | \( 1 + 4.71e16T + 6.49e32T^{2} \) |
| 3 | \( 1 + 1.38e26T + 1.01e52T^{2} \) |
| 5 | \( 1 + 1.39e38T + 1.54e76T^{2} \) |
| 7 | \( 1 - 1.59e46T + 1.30e92T^{2} \) |
| 11 | \( 1 + 5.93e56T + 3.24e113T^{2} \) |
| 13 | \( 1 - 1.89e60T + 2.62e121T^{2} \) |
| 17 | \( 1 + 1.15e67T + 1.31e134T^{2} \) |
| 19 | \( 1 + 3.88e69T + 2.42e139T^{2} \) |
| 23 | \( 1 + 1.96e74T + 2.68e148T^{2} \) |
| 29 | \( 1 - 5.20e79T + 2.51e159T^{2} \) |
| 31 | \( 1 - 8.91e80T + 3.61e162T^{2} \) |
| 37 | \( 1 - 1.37e85T + 8.58e170T^{2} \) |
| 41 | \( 1 - 8.15e87T + 6.21e175T^{2} \) |
| 43 | \( 1 - 3.10e88T + 1.11e178T^{2} \) |
| 47 | \( 1 + 1.80e91T + 1.81e182T^{2} \) |
| 53 | \( 1 - 1.31e94T + 8.83e187T^{2} \) |
| 59 | \( 1 + 5.81e95T + 1.05e193T^{2} \) |
| 61 | \( 1 + 2.73e97T + 3.98e194T^{2} \) |
| 67 | \( 1 - 1.37e99T + 1.10e199T^{2} \) |
| 71 | \( 1 + 9.18e99T + 6.12e201T^{2} \) |
| 73 | \( 1 + 1.17e101T + 1.26e203T^{2} \) |
| 79 | \( 1 - 4.74e103T + 6.93e206T^{2} \) |
| 83 | \( 1 - 7.42e104T + 1.51e209T^{2} \) |
| 89 | \( 1 + 1.07e106T + 3.04e212T^{2} \) |
| 97 | \( 1 - 2.33e108T + 3.61e216T^{2} \) |
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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
−11.98212406636930567443865544639, −11.17567350046476514431168872493, −10.51823649465999753367615228195, −8.419089219073201957793088056998, −7.69899032927477367834986104210, −6.26419075975595584579774196566, −4.60513163289099376473947601882, −2.16215318242575778828359585831, −0.805411289186455820329643643205, 0,
0.805411289186455820329643643205, 2.16215318242575778828359585831, 4.60513163289099376473947601882, 6.26419075975595584579774196566, 7.69899032927477367834986104210, 8.419089219073201957793088056998, 10.51823649465999753367615228195, 11.17567350046476514431168872493, 11.98212406636930567443865544639
