| L(s) = 1 | − 1.69e12·2-s + 1.74e19·3-s + 4.43e23·4-s − 9.34e27·5-s − 2.94e31·6-s + 1.25e34·7-s + 3.33e36·8-s − 1.40e38·9-s + 1.58e40·10-s − 1.22e42·11-s + 7.72e42·12-s − 2.40e44·13-s − 2.12e46·14-s − 1.62e47·15-s − 6.72e48·16-s + 5.23e49·17-s + 2.37e50·18-s + 4.30e51·19-s − 4.14e51·20-s + 2.18e53·21-s + 2.07e54·22-s + 2.46e55·23-s + 5.81e55·24-s − 3.26e56·25-s + 4.07e56·26-s − 1.01e58·27-s + 5.56e57·28-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 0.826·3-s + 0.183·4-s − 0.459·5-s − 0.899·6-s + 0.744·7-s + 0.888·8-s − 0.316·9-s + 0.499·10-s − 0.818·11-s + 0.151·12-s − 0.184·13-s − 0.810·14-s − 0.379·15-s − 1.14·16-s + 0.768·17-s + 0.344·18-s + 0.699·19-s − 0.0842·20-s + 0.615·21-s + 0.890·22-s + 1.74·23-s + 0.734·24-s − 0.788·25-s + 0.201·26-s − 1.08·27-s + 0.136·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(82-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+81/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(41)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{83}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 1.69e12T + 2.41e24T^{2} \) |
| 3 | \( 1 - 1.74e19T + 4.43e38T^{2} \) |
| 5 | \( 1 + 9.34e27T + 4.13e56T^{2} \) |
| 7 | \( 1 - 1.25e34T + 2.83e68T^{2} \) |
| 11 | \( 1 + 1.22e42T + 2.25e84T^{2} \) |
| 13 | \( 1 + 2.40e44T + 1.69e90T^{2} \) |
| 17 | \( 1 - 5.23e49T + 4.63e99T^{2} \) |
| 19 | \( 1 - 4.30e51T + 3.79e103T^{2} \) |
| 23 | \( 1 - 2.46e55T + 1.99e110T^{2} \) |
| 29 | \( 1 + 1.43e59T + 2.84e118T^{2} \) |
| 31 | \( 1 - 3.22e60T + 6.31e120T^{2} \) |
| 37 | \( 1 - 3.75e63T + 1.05e127T^{2} \) |
| 41 | \( 1 + 2.79e65T + 4.32e130T^{2} \) |
| 43 | \( 1 + 1.62e66T + 2.04e132T^{2} \) |
| 47 | \( 1 + 7.84e67T + 2.75e135T^{2} \) |
| 53 | \( 1 + 1.22e69T + 4.63e139T^{2} \) |
| 59 | \( 1 + 6.83e71T + 2.74e143T^{2} \) |
| 61 | \( 1 + 2.07e72T + 4.08e144T^{2} \) |
| 67 | \( 1 + 9.39e73T + 8.16e147T^{2} \) |
| 71 | \( 1 - 1.00e75T + 8.95e149T^{2} \) |
| 73 | \( 1 + 1.94e75T + 8.49e150T^{2} \) |
| 79 | \( 1 - 5.95e76T + 5.10e153T^{2} \) |
| 83 | \( 1 + 2.52e77T + 2.78e155T^{2} \) |
| 89 | \( 1 + 1.28e79T + 7.95e157T^{2} \) |
| 97 | \( 1 + 4.38e80T + 8.48e160T^{2} \) |
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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
−15.06910281221709824578636425849, −13.56276843286497901121979094140, −11.30848181160181212311971184637, −9.688051797242822105853457607311, −8.316239016575983458178636620600, −7.62608474807366610077595806406, −4.93941859638756811073626018852, −3.06980224411845227876692188932, −1.47106051999533347585562531642, 0,
1.47106051999533347585562531642, 3.06980224411845227876692188932, 4.93941859638756811073626018852, 7.62608474807366610077595806406, 8.316239016575983458178636620600, 9.688051797242822105853457607311, 11.30848181160181212311971184637, 13.56276843286497901121979094140, 15.06910281221709824578636425849
