| L(s) = 1 | + 8.79e12·2-s + 1.94e18·3-s + 7.73e25·4-s − 1.48e30·5-s + 1.70e31·6-s + 3.06e35·7-s + 6.80e38·8-s − 3.23e41·9-s − 1.30e43·10-s + 3.31e45·11-s + 1.50e44·12-s − 2.23e48·13-s + 2.69e48·14-s − 2.88e48·15-s + 5.98e51·16-s + 5.17e52·17-s − 2.84e54·18-s + 5.90e55·19-s − 1.14e56·20-s + 5.95e53·21-s + 2.91e58·22-s − 1.78e59·23-s + 1.32e57·24-s − 4.25e60·25-s − 1.96e61·26-s − 1.25e60·27-s + 2.37e61·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.00341·3-s + 0.5·4-s − 0.584·5-s + 0.00241·6-s + 0.0530·7-s + 0.353·8-s − 0.999·9-s − 0.413·10-s + 1.65·11-s + 0.00170·12-s − 0.782·13-s + 0.0375·14-s − 0.00199·15-s + 0.250·16-s + 0.154·17-s − 0.707·18-s + 1.39·19-s − 0.292·20-s + 0.000181·21-s + 1.17·22-s − 1.03·23-s + 0.00120·24-s − 0.658·25-s − 0.553·26-s − 0.00683·27-s + 0.0265·28-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 | \( 1 - 1.94e18T + 3.23e41T^{2} \) |
| 5 | \( 1 + 1.48e30T + 6.46e60T^{2} \) |
| 7 | \( 1 - 3.06e35T + 3.33e73T^{2} \) |
| 11 | \( 1 - 3.31e45T + 3.99e90T^{2} \) |
| 13 | \( 1 + 2.23e48T + 8.18e96T^{2} \) |
| 17 | \( 1 - 5.17e52T + 1.11e107T^{2} \) |
| 19 | \( 1 - 5.90e55T + 1.78e111T^{2} \) |
| 23 | \( 1 + 1.78e59T + 2.95e118T^{2} \) |
| 29 | \( 1 - 7.16e62T + 1.69e127T^{2} \) |
| 31 | \( 1 - 3.20e63T + 5.60e129T^{2} \) |
| 37 | \( 1 + 7.99e67T + 2.71e136T^{2} \) |
| 41 | \( 1 - 2.83e69T + 2.05e140T^{2} \) |
| 43 | \( 1 + 1.74e71T + 1.29e142T^{2} \) |
| 47 | \( 1 + 1.86e72T + 2.96e145T^{2} \) |
| 53 | \( 1 + 7.26e74T + 1.02e150T^{2} \) |
| 59 | \( 1 + 1.25e76T + 1.15e154T^{2} \) |
| 61 | \( 1 - 6.16e77T + 2.10e155T^{2} \) |
| 67 | \( 1 + 1.22e79T + 7.38e158T^{2} \) |
| 71 | \( 1 + 5.46e80T + 1.14e161T^{2} \) |
| 73 | \( 1 - 1.44e81T + 1.28e162T^{2} \) |
| 79 | \( 1 + 6.33e81T + 1.24e165T^{2} \) |
| 83 | \( 1 + 4.56e83T + 9.11e166T^{2} \) |
| 89 | \( 1 + 3.23e84T + 3.95e169T^{2} \) |
| 97 | \( 1 + 1.98e86T + 7.06e172T^{2} \) |
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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
−12.01483878130747954368909989026, −11.55537630380673684176511177223, −9.625110860310590459457723218420, −8.063930953941152305507304436573, −6.70692841807309708440926005677, −5.41926858206947779249228563708, −4.05267500699559602917734425279, −3.05890256000090809965131092828, −1.53114401983217974670993523428, 0,
1.53114401983217974670993523428, 3.05890256000090809965131092828, 4.05267500699559602917734425279, 5.41926858206947779249228563708, 6.70692841807309708440926005677, 8.063930953941152305507304436573, 9.625110860310590459457723218420, 11.55537630380673684176511177223, 12.01483878130747954368909989026
