| L(s) = 1 | + 8.79e12·2-s + 5.36e20·3-s + 7.73e25·4-s + 2.66e30·5-s + 4.71e33·6-s − 4.57e36·7-s + 6.80e38·8-s − 3.53e40·9-s + 2.34e43·10-s − 2.53e45·11-s + 4.15e46·12-s + 1.04e48·13-s − 4.02e49·14-s + 1.42e51·15-s + 5.98e51·16-s − 5.71e53·17-s − 3.11e53·18-s − 4.95e55·19-s + 2.05e56·20-s − 2.45e57·21-s − 2.22e58·22-s − 6.18e58·23-s + 3.65e59·24-s + 6.18e59·25-s + 9.20e60·26-s − 1.92e62·27-s − 3.53e62·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.943·3-s + 0.5·4-s + 1.04·5-s + 0.667·6-s − 0.791·7-s + 0.353·8-s − 0.109·9-s + 0.740·10-s − 1.26·11-s + 0.471·12-s + 0.365·13-s − 0.559·14-s + 0.987·15-s + 0.250·16-s − 1.70·17-s − 0.0774·18-s − 1.17·19-s + 0.523·20-s − 0.746·21-s − 0.895·22-s − 0.359·23-s + 0.333·24-s + 0.0957·25-s + 0.258·26-s − 1.04·27-s − 0.395·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(88-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+87/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(44)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{89}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8.79e12T \) |
| good | 3 | \( 1 - 5.36e20T + 3.23e41T^{2} \) |
| 5 | \( 1 - 2.66e30T + 6.46e60T^{2} \) |
| 7 | \( 1 + 4.57e36T + 3.33e73T^{2} \) |
| 11 | \( 1 + 2.53e45T + 3.99e90T^{2} \) |
| 13 | \( 1 - 1.04e48T + 8.18e96T^{2} \) |
| 17 | \( 1 + 5.71e53T + 1.11e107T^{2} \) |
| 19 | \( 1 + 4.95e55T + 1.78e111T^{2} \) |
| 23 | \( 1 + 6.18e58T + 2.95e118T^{2} \) |
| 29 | \( 1 + 3.05e63T + 1.69e127T^{2} \) |
| 31 | \( 1 - 1.19e64T + 5.60e129T^{2} \) |
| 37 | \( 1 - 2.02e67T + 2.71e136T^{2} \) |
| 41 | \( 1 - 1.29e70T + 2.05e140T^{2} \) |
| 43 | \( 1 - 9.63e70T + 1.29e142T^{2} \) |
| 47 | \( 1 + 8.51e72T + 2.96e145T^{2} \) |
| 53 | \( 1 - 1.31e75T + 1.02e150T^{2} \) |
| 59 | \( 1 - 2.06e77T + 1.15e154T^{2} \) |
| 61 | \( 1 - 2.23e77T + 2.10e155T^{2} \) |
| 67 | \( 1 - 2.06e79T + 7.38e158T^{2} \) |
| 71 | \( 1 - 4.05e80T + 1.14e161T^{2} \) |
| 73 | \( 1 + 1.20e81T + 1.28e162T^{2} \) |
| 79 | \( 1 - 2.21e82T + 1.24e165T^{2} \) |
| 83 | \( 1 + 1.22e83T + 9.11e166T^{2} \) |
| 89 | \( 1 - 1.03e85T + 3.95e169T^{2} \) |
| 97 | \( 1 + 3.26e86T + 7.06e172T^{2} \) |
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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
−13.08656383954524649859158619087, −10.89421195603292099594427927019, −9.565641323378641633569488577006, −8.302041294875529015128684269177, −6.61102519709581192739482148140, −5.55282662178783676115181858669, −3.96721285049306411628655174086, −2.57994939793480002216417989503, −2.13391203631453267541258132350, 0,
2.13391203631453267541258132350, 2.57994939793480002216417989503, 3.96721285049306411628655174086, 5.55282662178783676115181858669, 6.61102519709581192739482148140, 8.302041294875529015128684269177, 9.565641323378641633569488577006, 10.89421195603292099594427927019, 13.08656383954524649859158619087
