| L(s) = 1 | + 2.42·3-s + 2.44·5-s + 4.41·7-s + 2.89·9-s − 5.93·11-s + 4.07·13-s + 5.92·15-s + 2.12·17-s + 19-s + 10.7·21-s − 5.18·23-s + 0.962·25-s − 0.259·27-s + 2.70·29-s + 3.32·31-s − 14.3·33-s + 10.7·35-s − 7.04·37-s + 9.90·39-s − 0.996·41-s + 12.7·43-s + 7.06·45-s + 6.09·47-s + 12.5·49-s + 5.16·51-s + 53-s − 14.4·55-s + ⋯ |
| L(s) = 1 | + 1.40·3-s + 1.09·5-s + 1.66·7-s + 0.964·9-s − 1.78·11-s + 1.13·13-s + 1.53·15-s + 0.515·17-s + 0.229·19-s + 2.33·21-s − 1.08·23-s + 0.192·25-s − 0.0498·27-s + 0.503·29-s + 0.596·31-s − 2.50·33-s + 1.82·35-s − 1.15·37-s + 1.58·39-s − 0.155·41-s + 1.94·43-s + 1.05·45-s + 0.889·47-s + 1.78·49-s + 0.722·51-s + 0.137·53-s − 1.95·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.666998301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.666998301\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - 2.42T + 3T^{2} \) |
| 5 | \( 1 - 2.44T + 5T^{2} \) |
| 7 | \( 1 - 4.41T + 7T^{2} \) |
| 11 | \( 1 + 5.93T + 11T^{2} \) |
| 13 | \( 1 - 4.07T + 13T^{2} \) |
| 17 | \( 1 - 2.12T + 17T^{2} \) |
| 23 | \( 1 + 5.18T + 23T^{2} \) |
| 29 | \( 1 - 2.70T + 29T^{2} \) |
| 31 | \( 1 - 3.32T + 31T^{2} \) |
| 37 | \( 1 + 7.04T + 37T^{2} \) |
| 41 | \( 1 + 0.996T + 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 - 6.09T + 47T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 4.30T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 2.87T + 71T^{2} \) |
| 73 | \( 1 + 6.21T + 73T^{2} \) |
| 79 | \( 1 - 2.86T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 4.25T + 89T^{2} \) |
| 97 | \( 1 + 3.67T + 97T^{2} \) |
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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
−8.334855331279554987969353099210, −7.928149685136885222499387255886, −7.40597601067433109719431346221, −6.01529527002421906520729711597, −5.48798247637295005064674472921, −4.68769606302283344965502652508, −3.72400246793333903487753026192, −2.66360954358805605158243603769, −2.13775823851746463204216585076, −1.32015236922394849806488711188,
1.32015236922394849806488711188, 2.13775823851746463204216585076, 2.66360954358805605158243603769, 3.72400246793333903487753026192, 4.68769606302283344965502652508, 5.48798247637295005064674472921, 6.01529527002421906520729711597, 7.40597601067433109719431346221, 7.928149685136885222499387255886, 8.334855331279554987969353099210
