| L(s) = 1 | + 2-s − 0.252·3-s + 4-s + 2·5-s − 0.252·6-s + 1.74·7-s + 8-s − 2.93·9-s + 2·10-s + 11-s − 0.252·12-s + 2.93·13-s + 1.74·14-s − 0.504·15-s + 16-s − 1.68·17-s − 2.93·18-s + 2·20-s − 0.440·21-s + 22-s + 4.50·23-s − 0.252·24-s − 25-s + 2.93·26-s + 1.49·27-s + 1.74·28-s + 29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.145·3-s + 0.5·4-s + 0.894·5-s − 0.102·6-s + 0.660·7-s + 0.353·8-s − 0.978·9-s + 0.632·10-s + 0.301·11-s − 0.0727·12-s + 0.814·13-s + 0.467·14-s − 0.130·15-s + 0.250·16-s − 0.408·17-s − 0.692·18-s + 0.447·20-s − 0.0961·21-s + 0.213·22-s + 0.939·23-s − 0.0514·24-s − 0.200·25-s + 0.575·26-s + 0.287·27-s + 0.330·28-s + 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.192551509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.192551509\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.252T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 - 1.74T + 7T^{2} \) |
| 13 | \( 1 - 2.93T + 13T^{2} \) |
| 17 | \( 1 + 1.68T + 17T^{2} \) |
| 23 | \( 1 - 4.50T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + 5.43T + 31T^{2} \) |
| 37 | \( 1 - 3.18T + 37T^{2} \) |
| 41 | \( 1 - 2.18T + 41T^{2} \) |
| 43 | \( 1 - 7.18T + 43T^{2} \) |
| 47 | \( 1 - 6.62T + 47T^{2} \) |
| 53 | \( 1 - 1.00T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 9.44T + 61T^{2} \) |
| 67 | \( 1 - 0.756T + 67T^{2} \) |
| 71 | \( 1 - 7.18T + 71T^{2} \) |
| 73 | \( 1 - 8.18T + 73T^{2} \) |
| 79 | \( 1 + 2.75T + 79T^{2} \) |
| 83 | \( 1 + 3.69T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−7.79181358307515612594154347162, −6.93964138698285941390052658602, −6.25033800371169634573143725316, −5.67785334544387720613862663172, −5.22692778574317429611077468893, −4.34858909980298133229597318757, −3.56924868512559938229508658175, −2.63798853810012910525230211139, −1.95714779601577247844752340026, −0.950235893761683292244513349617,
0.950235893761683292244513349617, 1.95714779601577247844752340026, 2.63798853810012910525230211139, 3.56924868512559938229508658175, 4.34858909980298133229597318757, 5.22692778574317429611077468893, 5.67785334544387720613862663172, 6.25033800371169634573143725316, 6.93964138698285941390052658602, 7.79181358307515612594154347162
