| L(s) = 1 | − 6.88e11·2-s + 1.25e18·3-s + 3.22e23·4-s + 7.49e26·5-s − 8.65e29·6-s + 4.37e32·7-s − 1.18e35·8-s − 3.89e36·9-s − 5.15e38·10-s − 2.13e40·11-s + 4.05e41·12-s + 3.45e42·13-s − 3.00e44·14-s + 9.41e44·15-s + 3.26e46·16-s − 1.55e47·17-s + 2.68e48·18-s + 1.76e49·19-s + 2.41e50·20-s + 5.49e50·21-s + 1.46e52·22-s − 4.81e52·23-s − 1.48e53·24-s − 1.00e53·25-s − 2.38e54·26-s − 1.17e55·27-s + 1.41e56·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 0.537·3-s + 2.13·4-s + 0.920·5-s − 0.951·6-s + 1.27·7-s − 2.01·8-s − 0.711·9-s − 1.63·10-s − 1.71·11-s + 1.14·12-s + 0.448·13-s − 2.25·14-s + 0.494·15-s + 1.42·16-s − 0.660·17-s + 1.25·18-s + 1.03·19-s + 1.96·20-s + 0.683·21-s + 3.04·22-s − 1.80·23-s − 1.08·24-s − 0.152·25-s − 0.795·26-s − 0.919·27-s + 2.71·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(78-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+77/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(39)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{79}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 6.88e11T + 1.51e23T^{2} \) |
| 3 | \( 1 - 1.25e18T + 5.47e36T^{2} \) |
| 5 | \( 1 - 7.49e26T + 6.61e53T^{2} \) |
| 7 | \( 1 - 4.37e32T + 1.18e65T^{2} \) |
| 11 | \( 1 + 2.13e40T + 1.53e80T^{2} \) |
| 13 | \( 1 - 3.45e42T + 5.93e85T^{2} \) |
| 17 | \( 1 + 1.55e47T + 5.55e94T^{2} \) |
| 19 | \( 1 - 1.76e49T + 2.91e98T^{2} \) |
| 23 | \( 1 + 4.81e52T + 7.12e104T^{2} \) |
| 29 | \( 1 - 1.07e56T + 4.02e112T^{2} \) |
| 31 | \( 1 + 2.03e57T + 6.83e114T^{2} \) |
| 37 | \( 1 + 2.31e60T + 5.64e120T^{2} \) |
| 41 | \( 1 - 8.70e61T + 1.52e124T^{2} \) |
| 43 | \( 1 - 1.34e62T + 5.98e125T^{2} \) |
| 47 | \( 1 + 1.39e64T + 5.64e128T^{2} \) |
| 53 | \( 1 - 6.15e65T + 5.87e132T^{2} \) |
| 59 | \( 1 + 1.10e67T + 2.26e136T^{2} \) |
| 61 | \( 1 - 1.77e68T + 2.95e137T^{2} \) |
| 67 | \( 1 - 1.11e70T + 4.05e140T^{2} \) |
| 71 | \( 1 - 8.55e70T + 3.52e142T^{2} \) |
| 73 | \( 1 + 5.87e71T + 2.99e143T^{2} \) |
| 79 | \( 1 + 1.64e72T + 1.31e146T^{2} \) |
| 83 | \( 1 + 1.48e74T + 5.87e147T^{2} \) |
| 89 | \( 1 + 1.76e75T + 1.26e150T^{2} \) |
| 97 | \( 1 + 3.86e75T + 9.58e152T^{2} \) |
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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
−15.83213318430289590662369041964, −13.93609730798178147819336328681, −11.25380845784897394539774863743, −10.02096914130998063403588885052, −8.561302171911294709368530234230, −7.73359636451078555992886593182, −5.61672782074057470543426505704, −2.50795140872845410422367768551, −1.68083763095949248529949656486, 0,
1.68083763095949248529949656486, 2.50795140872845410422367768551, 5.61672782074057470543426505704, 7.73359636451078555992886593182, 8.561302171911294709368530234230, 10.02096914130998063403588885052, 11.25380845784897394539774863743, 13.93609730798178147819336328681, 15.83213318430289590662369041964
