| L(s) = 1 | + 2.41e12·2-s − 1.10e19·3-s + 3.41e24·4-s − 1.95e27·5-s − 2.66e31·6-s + 1.09e34·7-s + 2.42e36·8-s − 3.22e38·9-s − 4.72e39·10-s − 2.45e42·11-s − 3.76e43·12-s + 1.04e45·13-s + 2.64e46·14-s + 2.15e46·15-s − 2.42e48·16-s + 7.60e49·17-s − 7.78e50·18-s − 5.17e51·19-s − 6.68e51·20-s − 1.20e53·21-s − 5.93e54·22-s − 1.96e55·23-s − 2.66e55·24-s − 4.09e56·25-s + 2.53e57·26-s + 8.43e57·27-s + 3.74e58·28-s + ⋯ |
| L(s) = 1 | + 1.55·2-s − 0.523·3-s + 1.41·4-s − 0.0961·5-s − 0.812·6-s + 0.650·7-s + 0.643·8-s − 0.726·9-s − 0.149·10-s − 1.63·11-s − 0.739·12-s + 0.804·13-s + 1.01·14-s + 0.0503·15-s − 0.414·16-s + 1.11·17-s − 1.12·18-s − 0.839·19-s − 0.135·20-s − 0.340·21-s − 2.54·22-s − 1.38·23-s − 0.336·24-s − 0.990·25-s + 1.25·26-s + 0.903·27-s + 0.919·28-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 | \( 1 - 2.41e12T + 2.41e24T^{2} \) |
| 3 | \( 1 + 1.10e19T + 4.43e38T^{2} \) |
| 5 | \( 1 + 1.95e27T + 4.13e56T^{2} \) |
| 7 | \( 1 - 1.09e34T + 2.83e68T^{2} \) |
| 11 | \( 1 + 2.45e42T + 2.25e84T^{2} \) |
| 13 | \( 1 - 1.04e45T + 1.69e90T^{2} \) |
| 17 | \( 1 - 7.60e49T + 4.63e99T^{2} \) |
| 19 | \( 1 + 5.17e51T + 3.79e103T^{2} \) |
| 23 | \( 1 + 1.96e55T + 1.99e110T^{2} \) |
| 29 | \( 1 + 7.07e58T + 2.84e118T^{2} \) |
| 31 | \( 1 + 3.03e60T + 6.31e120T^{2} \) |
| 37 | \( 1 - 3.76e62T + 1.05e127T^{2} \) |
| 41 | \( 1 - 1.34e65T + 4.32e130T^{2} \) |
| 43 | \( 1 + 2.04e66T + 2.04e132T^{2} \) |
| 47 | \( 1 - 2.88e67T + 2.75e135T^{2} \) |
| 53 | \( 1 - 8.48e69T + 4.63e139T^{2} \) |
| 59 | \( 1 + 4.61e71T + 2.74e143T^{2} \) |
| 61 | \( 1 - 2.95e72T + 4.08e144T^{2} \) |
| 67 | \( 1 - 1.59e73T + 8.16e147T^{2} \) |
| 71 | \( 1 + 7.76e74T + 8.95e149T^{2} \) |
| 73 | \( 1 - 1.29e74T + 8.49e150T^{2} \) |
| 79 | \( 1 - 8.99e76T + 5.10e153T^{2} \) |
| 83 | \( 1 - 8.55e77T + 2.78e155T^{2} \) |
| 89 | \( 1 - 1.10e79T + 7.95e157T^{2} \) |
| 97 | \( 1 - 3.05e80T + 8.48e160T^{2} \) |
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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
−14.70809558177877896579605624890, −13.37726702706444157797515363524, −11.97922621034660380736282505732, −10.77848228390357631192064026055, −8.000352589725241571296479978893, −5.94339115901154556450448463112, −5.19028915907625799323707050125, −3.67982111958668632159476417666, −2.20083724671582977785592112581, 0,
2.20083724671582977785592112581, 3.67982111958668632159476417666, 5.19028915907625799323707050125, 5.94339115901154556450448463112, 8.000352589725241571296479978893, 10.77848228390357631192064026055, 11.97922621034660380736282505732, 13.37726702706444157797515363524, 14.70809558177877896579605624890
