| L(s) = 1 | + 1.79·5-s − 7-s + 3.83·11-s − 3.43·13-s + 0.293·17-s − 19-s + 6.83·23-s − 1.78·25-s + 5.62·29-s − 4.78·31-s − 1.79·35-s + 11.5·37-s − 5.57·41-s + 4.78·43-s − 12.7·47-s + 49-s + 4.12·53-s + 6.87·55-s + 11.5·61-s − 6.16·65-s + 10.2·67-s + 3.53·71-s − 10.4·73-s − 3.83·77-s − 2.22·79-s + 17.7·83-s + 0.526·85-s + ⋯ |
| L(s) = 1 | + 0.802·5-s − 0.377·7-s + 1.15·11-s − 0.953·13-s + 0.0711·17-s − 0.229·19-s + 1.42·23-s − 0.356·25-s + 1.04·29-s − 0.859·31-s − 0.303·35-s + 1.90·37-s − 0.871·41-s + 0.729·43-s − 1.85·47-s + 0.142·49-s + 0.566·53-s + 0.927·55-s + 1.48·61-s − 0.764·65-s + 1.24·67-s + 0.420·71-s − 1.22·73-s − 0.436·77-s − 0.249·79-s + 1.95·83-s + 0.0571·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.294672831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.294672831\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 1.79T + 5T^{2} \) |
| 11 | \( 1 - 3.83T + 11T^{2} \) |
| 13 | \( 1 + 3.43T + 13T^{2} \) |
| 17 | \( 1 - 0.293T + 17T^{2} \) |
| 23 | \( 1 - 6.83T + 23T^{2} \) |
| 29 | \( 1 - 5.62T + 29T^{2} \) |
| 31 | \( 1 + 4.78T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 + 5.57T + 41T^{2} \) |
| 43 | \( 1 - 4.78T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 4.12T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 3.53T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 2.22T + 79T^{2} \) |
| 83 | \( 1 - 17.7T + 83T^{2} \) |
| 89 | \( 1 - 5.08T + 89T^{2} \) |
| 97 | \( 1 - 9.52T + 97T^{2} \) |
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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
−8.382070942984535254732356641169, −7.43705186614841804902617808809, −6.71386643370478455615051921590, −6.25131841746072517554080176928, −5.35247709401873527083703035981, −4.65645194978473531664161392639, −3.72525572088040779686104705666, −2.80320590836861123465855947910, −1.95305875134974003450157961823, −0.852901522617359620717966458676,
0.852901522617359620717966458676, 1.95305875134974003450157961823, 2.80320590836861123465855947910, 3.72525572088040779686104705666, 4.65645194978473531664161392639, 5.35247709401873527083703035981, 6.25131841746072517554080176928, 6.71386643370478455615051921590, 7.43705186614841804902617808809, 8.382070942984535254732356641169
