| L(s) = 1 | + 3.08e15·2-s + 1.03e26·3-s − 6.39e32·4-s − 4.34e37·5-s + 3.19e41·6-s + 1.35e46·7-s − 3.97e48·8-s + 5.38e50·9-s − 1.34e53·10-s − 4.88e56·11-s − 6.60e58·12-s + 3.22e60·13-s + 4.19e61·14-s − 4.49e63·15-s + 4.02e65·16-s + 8.54e65·17-s + 1.66e66·18-s + 4.65e69·19-s + 2.77e70·20-s + 1.40e72·21-s − 1.50e72·22-s − 2.02e74·23-s − 4.11e74·24-s − 1.35e76·25-s + 9.95e75·26-s − 9.92e77·27-s − 8.67e78·28-s + ⋯ |
| L(s) = 1 | + 0.121·2-s + 1.02·3-s − 0.985·4-s − 0.350·5-s + 0.124·6-s + 1.18·7-s − 0.240·8-s + 0.0530·9-s − 0.0424·10-s − 0.856·11-s − 1.01·12-s + 0.628·13-s + 0.144·14-s − 0.359·15-s + 0.956·16-s + 0.0745·17-s + 0.00643·18-s + 0.945·19-s + 0.345·20-s + 1.21·21-s − 0.103·22-s − 1.23·23-s − 0.246·24-s − 0.877·25-s + 0.0762·26-s − 0.971·27-s − 1.17·28-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 | \( 1 - 3.08e15T + 6.49e32T^{2} \) |
| 3 | \( 1 - 1.03e26T + 1.01e52T^{2} \) |
| 5 | \( 1 + 4.34e37T + 1.54e76T^{2} \) |
| 7 | \( 1 - 1.35e46T + 1.30e92T^{2} \) |
| 11 | \( 1 + 4.88e56T + 3.24e113T^{2} \) |
| 13 | \( 1 - 3.22e60T + 2.62e121T^{2} \) |
| 17 | \( 1 - 8.54e65T + 1.31e134T^{2} \) |
| 19 | \( 1 - 4.65e69T + 2.42e139T^{2} \) |
| 23 | \( 1 + 2.02e74T + 2.68e148T^{2} \) |
| 29 | \( 1 + 1.16e79T + 2.51e159T^{2} \) |
| 31 | \( 1 - 3.19e81T + 3.61e162T^{2} \) |
| 37 | \( 1 + 1.89e84T + 8.58e170T^{2} \) |
| 41 | \( 1 + 1.33e88T + 6.21e175T^{2} \) |
| 43 | \( 1 - 2.15e88T + 1.11e178T^{2} \) |
| 47 | \( 1 + 1.66e91T + 1.81e182T^{2} \) |
| 53 | \( 1 - 5.64e92T + 8.83e187T^{2} \) |
| 59 | \( 1 + 4.75e96T + 1.05e193T^{2} \) |
| 61 | \( 1 - 1.23e97T + 3.98e194T^{2} \) |
| 67 | \( 1 - 4.08e99T + 1.10e199T^{2} \) |
| 71 | \( 1 + 4.74e100T + 6.12e201T^{2} \) |
| 73 | \( 1 + 9.49e100T + 1.26e203T^{2} \) |
| 79 | \( 1 + 4.54e103T + 6.93e206T^{2} \) |
| 83 | \( 1 + 6.03e104T + 1.51e209T^{2} \) |
| 89 | \( 1 + 1.59e106T + 3.04e212T^{2} \) |
| 97 | \( 1 - 2.56e108T + 3.61e216T^{2} \) |
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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
−13.50219814760423591957017472780, −11.68488726226920895777071084257, −9.882563652995913170071145127497, −8.345321518948748900596273058633, −7.972440039866901589510052889325, −5.47537121896131193609298150744, −4.24471205543454593613266188902, −3.07444307082402688070990666526, −1.56553757023145886641107294414, 0,
1.56553757023145886641107294414, 3.07444307082402688070990666526, 4.24471205543454593613266188902, 5.47537121896131193609298150744, 7.972440039866901589510052889325, 8.345321518948748900596273058633, 9.882563652995913170071145127497, 11.68488726226920895777071084257, 13.50219814760423591957017472780
