| L(s) = 1 | + 3.43e10·2-s − 9.23e16·3-s + 1.18e21·4-s + 8.49e24·5-s − 3.17e27·6-s − 9.92e29·7-s + 4.05e31·8-s + 1.02e33·9-s + 2.92e35·10-s + 7.28e36·11-s − 1.09e38·12-s − 2.66e39·13-s − 3.41e40·14-s − 7.85e41·15-s + 1.39e42·16-s + 3.09e43·17-s + 3.51e43·18-s − 3.31e45·19-s + 1.00e46·20-s + 9.16e46·21-s + 2.50e47·22-s + 3.26e48·23-s − 3.74e48·24-s + 2.98e49·25-s − 9.15e49·26-s + 5.99e50·27-s − 1.17e51·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.06·3-s + 0.5·4-s + 1.30·5-s − 0.753·6-s − 0.990·7-s + 0.353·8-s + 0.136·9-s + 0.923·10-s + 0.781·11-s − 0.532·12-s − 0.759·13-s − 0.700·14-s − 1.39·15-s + 0.250·16-s + 0.645·17-s + 0.0962·18-s − 1.33·19-s + 0.652·20-s + 1.05·21-s + 0.552·22-s + 1.48·23-s − 0.376·24-s + 0.705·25-s − 0.537·26-s + 0.920·27-s − 0.495·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(72-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+71/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(36)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{73}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 3.43e10T \) |
| good | 3 | \( 1 + 9.23e16T + 7.50e33T^{2} \) |
| 5 | \( 1 - 8.49e24T + 4.23e49T^{2} \) |
| 7 | \( 1 + 9.92e29T + 1.00e60T^{2} \) |
| 11 | \( 1 - 7.28e36T + 8.68e73T^{2} \) |
| 13 | \( 1 + 2.66e39T + 1.23e79T^{2} \) |
| 17 | \( 1 - 3.09e43T + 2.30e87T^{2} \) |
| 19 | \( 1 + 3.31e45T + 6.18e90T^{2} \) |
| 23 | \( 1 - 3.26e48T + 4.81e96T^{2} \) |
| 29 | \( 1 + 1.26e52T + 6.76e103T^{2} \) |
| 31 | \( 1 - 8.89e52T + 7.70e105T^{2} \) |
| 37 | \( 1 + 6.86e55T + 2.19e111T^{2} \) |
| 41 | \( 1 - 1.41e57T + 3.21e114T^{2} \) |
| 43 | \( 1 + 1.16e58T + 9.46e115T^{2} \) |
| 47 | \( 1 + 2.61e59T + 5.23e118T^{2} \) |
| 53 | \( 1 + 1.32e61T + 2.65e122T^{2} \) |
| 59 | \( 1 + 6.93e62T + 5.37e125T^{2} \) |
| 61 | \( 1 + 3.89e63T + 5.73e126T^{2} \) |
| 67 | \( 1 + 4.41e64T + 4.48e129T^{2} \) |
| 71 | \( 1 + 5.78e65T + 2.75e131T^{2} \) |
| 73 | \( 1 + 6.21e65T + 1.97e132T^{2} \) |
| 79 | \( 1 - 4.06e67T + 5.38e134T^{2} \) |
| 83 | \( 1 + 1.39e67T + 1.79e136T^{2} \) |
| 89 | \( 1 - 1.62e69T + 2.55e138T^{2} \) |
| 97 | \( 1 - 2.89e69T + 1.15e141T^{2} \) |
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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
−13.26511680331823975075994087424, −12.19080807413699445482972019936, −10.66117128936890026361227093406, −9.405731504816053111406828769613, −6.70500597544973955191850689069, −6.00417733997043696939286395043, −4.90282610212446814134196825372, −3.07744928727549785355585396201, −1.60487348848149593570954337502, 0,
1.60487348848149593570954337502, 3.07744928727549785355585396201, 4.90282610212446814134196825372, 6.00417733997043696939286395043, 6.70500597544973955191850689069, 9.405731504816053111406828769613, 10.66117128936890026361227093406, 12.19080807413699445482972019936, 13.26511680331823975075994087424
