| L(s) = 1 | − 3.43e10·2-s + 6.35e16·3-s + 1.18e21·4-s + 1.19e25·5-s − 2.18e27·6-s − 1.27e30·7-s − 4.05e31·8-s − 3.47e33·9-s − 4.11e35·10-s − 1.31e35·11-s + 7.50e37·12-s + 4.90e39·13-s + 4.36e40·14-s + 7.61e41·15-s + 1.39e42·16-s − 6.48e43·17-s + 1.19e44·18-s + 2.72e45·19-s + 1.41e46·20-s − 8.07e46·21-s + 4.50e45·22-s + 2.34e48·23-s − 2.57e48·24-s + 1.01e50·25-s − 1.68e50·26-s − 6.97e50·27-s − 1.50e51·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.733·3-s + 0.5·4-s + 1.84·5-s − 0.518·6-s − 1.26·7-s − 0.353·8-s − 0.462·9-s − 1.30·10-s − 0.0140·11-s + 0.366·12-s + 1.39·13-s + 0.896·14-s + 1.34·15-s + 0.250·16-s − 1.35·17-s + 0.327·18-s + 1.09·19-s + 0.920·20-s − 0.929·21-s + 0.00995·22-s + 1.06·23-s − 0.259·24-s + 2.38·25-s − 0.989·26-s − 1.07·27-s − 0.634·28-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
| \(L(36)\) |
\(\approx\) |
\(2.555598990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.555598990\) |
| \(L(\frac{73}{2})\) |
|
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 | \( 1 - 6.35e16T + 7.50e33T^{2} \) |
| 5 | \( 1 - 1.19e25T + 4.23e49T^{2} \) |
| 7 | \( 1 + 1.27e30T + 1.00e60T^{2} \) |
| 11 | \( 1 + 1.31e35T + 8.68e73T^{2} \) |
| 13 | \( 1 - 4.90e39T + 1.23e79T^{2} \) |
| 17 | \( 1 + 6.48e43T + 2.30e87T^{2} \) |
| 19 | \( 1 - 2.72e45T + 6.18e90T^{2} \) |
| 23 | \( 1 - 2.34e48T + 4.81e96T^{2} \) |
| 29 | \( 1 - 9.02e51T + 6.76e103T^{2} \) |
| 31 | \( 1 + 4.95e52T + 7.70e105T^{2} \) |
| 37 | \( 1 + 1.80e55T + 2.19e111T^{2} \) |
| 41 | \( 1 - 6.04e56T + 3.21e114T^{2} \) |
| 43 | \( 1 - 1.03e58T + 9.46e115T^{2} \) |
| 47 | \( 1 - 2.53e59T + 5.23e118T^{2} \) |
| 53 | \( 1 - 6.53e60T + 2.65e122T^{2} \) |
| 59 | \( 1 - 2.34e62T + 5.37e125T^{2} \) |
| 61 | \( 1 + 2.75e63T + 5.73e126T^{2} \) |
| 67 | \( 1 - 6.97e64T + 4.48e129T^{2} \) |
| 71 | \( 1 - 5.81e65T + 2.75e131T^{2} \) |
| 73 | \( 1 + 8.45e65T + 1.97e132T^{2} \) |
| 79 | \( 1 - 1.96e67T + 5.38e134T^{2} \) |
| 83 | \( 1 + 1.11e68T + 1.79e136T^{2} \) |
| 89 | \( 1 - 5.42e68T + 2.55e138T^{2} \) |
| 97 | \( 1 - 2.87e70T + 1.15e141T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.94182173861024884076158528876, −13.12608148517861575136317542174, −10.73768234587935097249065419450, −9.390456928277208564879103718363, −8.861465666356790097821213590195, −6.70007701866108697943991357577, −5.75985300194031720340061788654, −3.17556865280986370615013316643, −2.27551970500632059377150467691, −0.932287180389301422681322737017,
0.932287180389301422681322737017, 2.27551970500632059377150467691, 3.17556865280986370615013316643, 5.75985300194031720340061788654, 6.70007701866108697943991357577, 8.861465666356790097821213590195, 9.390456928277208564879103718363, 10.73768234587935097249065419450, 13.12608148517861575136317542174, 13.94182173861024884076158528876
