| L(s) = 1 | − 8.79e12·2-s − 3.08e20·3-s + 7.73e25·4-s + 1.64e30·5-s + 2.71e33·6-s − 1.87e36·7-s − 6.80e38·8-s − 2.28e41·9-s − 1.44e43·10-s − 1.23e45·11-s − 2.38e46·12-s − 1.24e48·13-s + 1.65e49·14-s − 5.08e50·15-s + 5.98e51·16-s + 1.26e53·17-s + 2.00e54·18-s + 8.44e54·19-s + 1.27e56·20-s + 5.79e56·21-s + 1.08e58·22-s + 7.04e58·23-s + 2.09e59·24-s − 3.74e60·25-s + 1.09e61·26-s + 1.70e62·27-s − 1.45e62·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.542·3-s + 0.5·4-s + 0.648·5-s + 0.383·6-s − 0.325·7-s − 0.353·8-s − 0.705·9-s − 0.458·10-s − 0.616·11-s − 0.271·12-s − 0.435·13-s + 0.229·14-s − 0.351·15-s + 0.250·16-s + 0.379·17-s + 0.499·18-s + 0.199·19-s + 0.324·20-s + 0.176·21-s + 0.436·22-s + 0.409·23-s + 0.191·24-s − 0.579·25-s + 0.308·26-s + 0.925·27-s − 0.162·28-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
| \(L(44)\) |
\(\approx\) |
\(0.7265179022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7265179022\) |
| \(L(\frac{89}{2})\) |
|
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 | \( 1 + 3.08e20T + 3.23e41T^{2} \) |
| 5 | \( 1 - 1.64e30T + 6.46e60T^{2} \) |
| 7 | \( 1 + 1.87e36T + 3.33e73T^{2} \) |
| 11 | \( 1 + 1.23e45T + 3.99e90T^{2} \) |
| 13 | \( 1 + 1.24e48T + 8.18e96T^{2} \) |
| 17 | \( 1 - 1.26e53T + 1.11e107T^{2} \) |
| 19 | \( 1 - 8.44e54T + 1.78e111T^{2} \) |
| 23 | \( 1 - 7.04e58T + 2.95e118T^{2} \) |
| 29 | \( 1 - 3.70e62T + 1.69e127T^{2} \) |
| 31 | \( 1 + 1.30e65T + 5.60e129T^{2} \) |
| 37 | \( 1 + 7.32e67T + 2.71e136T^{2} \) |
| 41 | \( 1 - 1.01e70T + 2.05e140T^{2} \) |
| 43 | \( 1 - 6.22e70T + 1.29e142T^{2} \) |
| 47 | \( 1 + 4.55e72T + 2.96e145T^{2} \) |
| 53 | \( 1 + 9.26e74T + 1.02e150T^{2} \) |
| 59 | \( 1 + 7.26e75T + 1.15e154T^{2} \) |
| 61 | \( 1 - 3.50e77T + 2.10e155T^{2} \) |
| 67 | \( 1 + 1.42e79T + 7.38e158T^{2} \) |
| 71 | \( 1 + 1.26e80T + 1.14e161T^{2} \) |
| 73 | \( 1 + 1.17e81T + 1.28e162T^{2} \) |
| 79 | \( 1 - 4.54e82T + 1.24e165T^{2} \) |
| 83 | \( 1 + 2.90e83T + 9.11e166T^{2} \) |
| 89 | \( 1 - 7.03e84T + 3.95e169T^{2} \) |
| 97 | \( 1 - 3.12e85T + 7.06e172T^{2} \) |
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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
−12.80901384142495335017123858665, −11.39091323204960910379151594629, −10.22110836950879164733897526352, −9.083263485578961601218456632233, −7.59075105648841395702588976167, −6.16954090348003784561976396902, −5.21678181675835808610135081145, −3.12392282633690103654311652179, −1.90737593458329548511964222685, −0.45834592509668309188427641053,
0.45834592509668309188427641053, 1.90737593458329548511964222685, 3.12392282633690103654311652179, 5.21678181675835808610135081145, 6.16954090348003784561976396902, 7.59075105648841395702588976167, 9.083263485578961601218456632233, 10.22110836950879164733897526352, 11.39091323204960910379151594629, 12.80901384142495335017123858665
