| L(s) = 1 | − 2.46·2-s − 2.68·3-s + 4.05·4-s + 2.96·5-s + 6.59·6-s + 0.509·7-s − 5.06·8-s + 4.19·9-s − 7.29·10-s + 5.31·11-s − 10.8·12-s + 5.60·13-s − 1.25·14-s − 7.94·15-s + 4.34·16-s + 2.47·17-s − 10.3·18-s − 0.319·19-s + 12.0·20-s − 1.36·21-s − 13.0·22-s + 2.93·23-s + 13.5·24-s + 3.78·25-s − 13.7·26-s − 3.19·27-s + 2.06·28-s + ⋯ |
| L(s) = 1 | − 1.74·2-s − 1.54·3-s + 2.02·4-s + 1.32·5-s + 2.69·6-s + 0.192·7-s − 1.78·8-s + 1.39·9-s − 2.30·10-s + 1.60·11-s − 3.14·12-s + 1.55·13-s − 0.335·14-s − 2.05·15-s + 1.08·16-s + 0.599·17-s − 2.43·18-s − 0.0733·19-s + 2.68·20-s − 0.298·21-s − 2.78·22-s + 0.612·23-s + 2.77·24-s + 0.757·25-s − 2.70·26-s − 0.615·27-s + 0.390·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9633387543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9633387543\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5077 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 3 | \( 1 + 2.68T + 3T^{2} \) |
| 5 | \( 1 - 2.96T + 5T^{2} \) |
| 7 | \( 1 - 0.509T + 7T^{2} \) |
| 11 | \( 1 - 5.31T + 11T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 + 0.319T + 19T^{2} \) |
| 23 | \( 1 - 2.93T + 23T^{2} \) |
| 29 | \( 1 + 4.69T + 29T^{2} \) |
| 31 | \( 1 - 7.77T + 31T^{2} \) |
| 37 | \( 1 - 9.69T + 37T^{2} \) |
| 41 | \( 1 + 4.12T + 41T^{2} \) |
| 43 | \( 1 - 2.69T + 43T^{2} \) |
| 47 | \( 1 - 1.32T + 47T^{2} \) |
| 53 | \( 1 - 3.72T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 2.43T + 61T^{2} \) |
| 67 | \( 1 - 9.21T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 6.94T + 73T^{2} \) |
| 79 | \( 1 + 9.50T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 2.74T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.501479692071198381886509087975, −7.47191159182041294530245936868, −6.58014471180274492247370583491, −6.27231053676058098614851704361, −5.84322342804376504133439621501, −4.83037472368913930641247738800, −3.66164566187486925424122275309, −2.20342573806860051277105059413, −1.16680563723700876545470665618, −1.00851312280412574868396732162,
1.00851312280412574868396732162, 1.16680563723700876545470665618, 2.20342573806860051277105059413, 3.66164566187486925424122275309, 4.83037472368913930641247738800, 5.84322342804376504133439621501, 6.27231053676058098614851704361, 6.58014471180274492247370583491, 7.47191159182041294530245936868, 8.501479692071198381886509087975
