L(s) = 1 | − 2.33·3-s − 5-s + 7-s + 2.43·9-s − 11-s + 1.04·13-s + 2.33·15-s + 6.88·17-s + 6.55·19-s − 2.33·21-s + 0.836·23-s + 25-s + 1.31·27-s + 8.55·29-s − 3.19·31-s + 2.33·33-s − 35-s + 7.50·37-s − 2.42·39-s − 7.72·41-s − 10.8·43-s − 2.43·45-s − 13.5·47-s + 49-s − 16.0·51-s + 3.85·53-s + 55-s + ⋯ |
L(s) = 1 | − 1.34·3-s − 0.447·5-s + 0.377·7-s + 0.812·9-s − 0.301·11-s + 0.288·13-s + 0.602·15-s + 1.66·17-s + 1.50·19-s − 0.508·21-s + 0.174·23-s + 0.200·25-s + 0.252·27-s + 1.58·29-s − 0.573·31-s + 0.405·33-s − 0.169·35-s + 1.23·37-s − 0.388·39-s − 1.20·41-s − 1.65·43-s − 0.363·45-s − 1.97·47-s + 0.142·49-s − 2.24·51-s + 0.529·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.213271132\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.213271132\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 2.33T + 3T^{2} \) |
| 13 | \( 1 - 1.04T + 13T^{2} \) |
| 17 | \( 1 - 6.88T + 17T^{2} \) |
| 19 | \( 1 - 6.55T + 19T^{2} \) |
| 23 | \( 1 - 0.836T + 23T^{2} \) |
| 29 | \( 1 - 8.55T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 - 7.50T + 37T^{2} \) |
| 41 | \( 1 + 7.72T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 13.5T + 47T^{2} \) |
| 53 | \( 1 - 3.85T + 53T^{2} \) |
| 59 | \( 1 - 7.16T + 59T^{2} \) |
| 61 | \( 1 - 2.29T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 3.89T + 71T^{2} \) |
| 73 | \( 1 - 0.547T + 73T^{2} \) |
| 79 | \( 1 + 8.83T + 79T^{2} \) |
| 83 | \( 1 - 7.89T + 83T^{2} \) |
| 89 | \( 1 - 7.67T + 89T^{2} \) |
| 97 | \( 1 + 6.52T + 97T^{2} \) |
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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
−8.061798042075309411144343937982, −7.24185939240720232601293911943, −6.61434693901353991961352474058, −5.80299350930823850759987489724, −5.14564935639273243247819882096, −4.82794739163985056249533513630, −3.61902566473755762057437581987, −2.96948596917922441156876145674, −1.42598465907299667994018195079, −0.68644876128711109824051294883,
0.68644876128711109824051294883, 1.42598465907299667994018195079, 2.96948596917922441156876145674, 3.61902566473755762057437581987, 4.82794739163985056249533513630, 5.14564935639273243247819882096, 5.80299350930823850759987489724, 6.61434693901353991961352474058, 7.24185939240720232601293911943, 8.061798042075309411144343937982