| L(s) = 1 | − 1.02e5·2-s + 8.77e7·3-s + 1.90e9·4-s − 8.98e12·6-s − 1.11e14·7-s + 6.84e14·8-s + 2.13e15·9-s − 2.24e17·11-s + 1.67e17·12-s − 2.46e18·13-s + 1.14e19·14-s − 8.65e19·16-s + 3.31e19·17-s − 2.19e20·18-s − 7.93e19·19-s − 9.81e21·21-s + 2.30e22·22-s − 2.88e22·23-s + 6.00e22·24-s + 2.52e23·26-s − 3.00e23·27-s − 2.13e23·28-s − 2.32e24·29-s − 7.38e24·31-s + 2.98e24·32-s − 1.97e25·33-s − 3.40e24·34-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 1.17·3-s + 0.222·4-s − 1.30·6-s − 1.27·7-s + 0.859·8-s + 0.384·9-s − 1.47·11-s + 0.261·12-s − 1.02·13-s + 1.40·14-s − 1.17·16-s + 0.165·17-s − 0.425·18-s − 0.0631·19-s − 1.49·21-s + 1.63·22-s − 0.979·23-s + 1.01·24-s + 1.13·26-s − 0.724·27-s − 0.282·28-s − 1.72·29-s − 1.82·31-s + 0.436·32-s − 1.73·33-s − 0.182·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(\approx\) |
\(0.1129099522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1129099522\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 1.02e5T + 8.58e9T^{2} \) |
| 3 | \( 1 - 8.77e7T + 5.55e15T^{2} \) |
| 7 | \( 1 + 1.11e14T + 7.73e27T^{2} \) |
| 11 | \( 1 + 2.24e17T + 2.32e34T^{2} \) |
| 13 | \( 1 + 2.46e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 3.31e19T + 4.02e40T^{2} \) |
| 19 | \( 1 + 7.93e19T + 1.58e42T^{2} \) |
| 23 | \( 1 + 2.88e22T + 8.65e44T^{2} \) |
| 29 | \( 1 + 2.32e24T + 1.81e48T^{2} \) |
| 31 | \( 1 + 7.38e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 1.63e25T + 5.63e51T^{2} \) |
| 41 | \( 1 - 4.37e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 2.93e26T + 8.02e53T^{2} \) |
| 47 | \( 1 - 6.15e26T + 1.51e55T^{2} \) |
| 53 | \( 1 - 1.66e28T + 7.96e56T^{2} \) |
| 59 | \( 1 + 6.84e28T + 2.74e58T^{2} \) |
| 61 | \( 1 + 3.85e29T + 8.23e58T^{2} \) |
| 67 | \( 1 - 1.10e30T + 1.82e60T^{2} \) |
| 71 | \( 1 - 7.46e29T + 1.23e61T^{2} \) |
| 73 | \( 1 + 6.29e30T + 3.08e61T^{2} \) |
| 79 | \( 1 - 1.61e31T + 4.18e62T^{2} \) |
| 83 | \( 1 + 2.30e31T + 2.13e63T^{2} \) |
| 89 | \( 1 - 1.08e32T + 2.13e64T^{2} \) |
| 97 | \( 1 + 2.96e32T + 3.65e65T^{2} \) |
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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
−10.72166170971476920485298049460, −9.674988110367767640963876476484, −9.184491830927116669969935469669, −7.895499252247895276042061041621, −7.35634237192121535843502835982, −5.54485732221752953873875933612, −3.90404331948102886234041266407, −2.75059112659339009029961236817, −1.95561477459474355328595865708, −0.15037520746959966461637356546,
0.15037520746959966461637356546, 1.95561477459474355328595865708, 2.75059112659339009029961236817, 3.90404331948102886234041266407, 5.54485732221752953873875933612, 7.35634237192121535843502835982, 7.895499252247895276042061041621, 9.184491830927116669969935469669, 9.674988110367767640963876476484, 10.72166170971476920485298049460
