| L(s) = 1 | − 1.25·2-s + 2.77·3-s − 0.422·4-s − 3.18·5-s − 3.48·6-s + 3.63·7-s + 3.04·8-s + 4.67·9-s + 3.99·10-s + 2.33·11-s − 1.17·12-s − 13-s − 4.56·14-s − 8.81·15-s − 2.97·16-s − 1.15·17-s − 5.87·18-s + 0.749·19-s + 1.34·20-s + 10.0·21-s − 2.92·22-s + 3.47·23-s + 8.43·24-s + 5.12·25-s + 1.25·26-s + 4.65·27-s − 1.53·28-s + ⋯ |
| L(s) = 1 | − 0.888·2-s + 1.59·3-s − 0.211·4-s − 1.42·5-s − 1.42·6-s + 1.37·7-s + 1.07·8-s + 1.55·9-s + 1.26·10-s + 0.703·11-s − 0.338·12-s − 0.277·13-s − 1.22·14-s − 2.27·15-s − 0.744·16-s − 0.279·17-s − 1.38·18-s + 0.171·19-s + 0.300·20-s + 2.19·21-s − 0.624·22-s + 0.724·23-s + 1.72·24-s + 1.02·25-s + 0.246·26-s + 0.895·27-s − 0.290·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.577929075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.577929075\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 1.25T + 2T^{2} \) |
| 3 | \( 1 - 2.77T + 3T^{2} \) |
| 5 | \( 1 + 3.18T + 5T^{2} \) |
| 7 | \( 1 - 3.63T + 7T^{2} \) |
| 11 | \( 1 - 2.33T + 11T^{2} \) |
| 17 | \( 1 + 1.15T + 17T^{2} \) |
| 19 | \( 1 - 0.749T + 19T^{2} \) |
| 23 | \( 1 - 3.47T + 23T^{2} \) |
| 29 | \( 1 - 8.01T + 29T^{2} \) |
| 31 | \( 1 + 5.92T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 5.56T + 41T^{2} \) |
| 43 | \( 1 - 0.0136T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 8.86T + 53T^{2} \) |
| 59 | \( 1 - 0.498T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 9.20T + 67T^{2} \) |
| 71 | \( 1 - 5.49T + 71T^{2} \) |
| 73 | \( 1 + 4.39T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 4.87T + 83T^{2} \) |
| 89 | \( 1 + 8.79T + 89T^{2} \) |
| 97 | \( 1 - 0.309T + 97T^{2} \) |
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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
−9.106144533511400456936859576448, −8.727356319023812289803614014221, −8.243402246972317522338691356497, −7.46901337154987189023484190936, −7.15755979969169168443883683226, −5.01342099889374413351483912799, −4.23229317975921113188916628261, −3.60546167272111000737320019344, −2.23069292483014675004199059498, −1.05188424335029828388311141375,
1.05188424335029828388311141375, 2.23069292483014675004199059498, 3.60546167272111000737320019344, 4.23229317975921113188916628261, 5.01342099889374413351483912799, 7.15755979969169168443883683226, 7.46901337154987189023484190936, 8.243402246972317522338691356497, 8.727356319023812289803614014221, 9.106144533511400456936859576448
