| L(s) = 1 | − 5.04e4·2-s − 3.54e7·3-s + 3.99e8·4-s + 1.78e12·6-s − 7.37e12·7-s + 8.82e13·8-s + 6.36e14·9-s − 2.56e16·11-s − 1.41e16·12-s − 1.59e17·13-s + 3.72e17·14-s − 5.31e18·16-s + 5.12e18·17-s − 3.21e19·18-s + 6.46e19·19-s + 2.61e20·21-s + 1.29e21·22-s + 1.49e21·23-s − 3.12e21·24-s + 8.07e21·26-s − 6.77e20·27-s − 2.94e21·28-s − 5.23e22·29-s + 2.77e22·31-s + 7.85e22·32-s + 9.09e23·33-s − 2.58e23·34-s + ⋯ |
| L(s) = 1 | − 1.08·2-s − 1.42·3-s + 0.186·4-s + 1.55·6-s − 0.587·7-s + 0.886·8-s + 1.03·9-s − 1.85·11-s − 0.265·12-s − 0.866·13-s + 0.639·14-s − 1.15·16-s + 0.434·17-s − 1.12·18-s + 0.977·19-s + 0.837·21-s + 2.01·22-s + 1.16·23-s − 1.26·24-s + 0.943·26-s − 0.0441·27-s − 0.109·28-s − 1.12·29-s + 0.212·31-s + 0.367·32-s + 2.64·33-s − 0.473·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+31/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(16)\) |
\(\approx\) |
\(0.007911832599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.007911832599\) |
| \(L(\frac{33}{2})\) |
|
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 | \( 1 + 5.04e4T + 2.14e9T^{2} \) |
| 3 | \( 1 + 3.54e7T + 6.17e14T^{2} \) |
| 7 | \( 1 + 7.37e12T + 1.57e26T^{2} \) |
| 11 | \( 1 + 2.56e16T + 1.91e32T^{2} \) |
| 13 | \( 1 + 1.59e17T + 3.40e34T^{2} \) |
| 17 | \( 1 - 5.12e18T + 1.39e38T^{2} \) |
| 19 | \( 1 - 6.46e19T + 4.37e39T^{2} \) |
| 23 | \( 1 - 1.49e21T + 1.63e42T^{2} \) |
| 29 | \( 1 + 5.23e22T + 2.15e45T^{2} \) |
| 31 | \( 1 - 2.77e22T + 1.70e46T^{2} \) |
| 37 | \( 1 - 6.74e23T + 4.11e48T^{2} \) |
| 41 | \( 1 + 6.66e24T + 9.91e49T^{2} \) |
| 43 | \( 1 + 2.14e25T + 4.34e50T^{2} \) |
| 47 | \( 1 - 1.75e25T + 6.83e51T^{2} \) |
| 53 | \( 1 - 3.62e26T + 2.83e53T^{2} \) |
| 59 | \( 1 + 4.99e27T + 7.87e54T^{2} \) |
| 61 | \( 1 + 7.77e27T + 2.21e55T^{2} \) |
| 67 | \( 1 + 2.47e28T + 4.05e56T^{2} \) |
| 71 | \( 1 - 8.55e28T + 2.44e57T^{2} \) |
| 73 | \( 1 - 7.99e28T + 5.79e57T^{2} \) |
| 79 | \( 1 + 1.78e29T + 6.70e58T^{2} \) |
| 83 | \( 1 + 8.73e29T + 3.10e59T^{2} \) |
| 89 | \( 1 + 2.53e30T + 2.69e60T^{2} \) |
| 97 | \( 1 - 2.82e30T + 3.88e61T^{2} \) |
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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
−11.13476459667242828728264301531, −10.30890822509143612154318646635, −9.494608891468205857506047917391, −7.889645043145821304655657935188, −7.01372068279006475817724694294, −5.49429040772845926480951830146, −4.84508474301317990399963268405, −2.88465749121369021797270984347, −1.29116332664020145396105326559, −0.05543699094028381186433910448,
0.05543699094028381186433910448, 1.29116332664020145396105326559, 2.88465749121369021797270984347, 4.84508474301317990399963268405, 5.49429040772845926480951830146, 7.01372068279006475817724694294, 7.889645043145821304655657935188, 9.494608891468205857506047917391, 10.30890822509143612154318646635, 11.13476459667242828728264301531
