| L(s) = 1 | + 2.66·2-s − 1.48·3-s + 5.09·4-s − 3.95·6-s + 2.27·7-s + 8.23·8-s − 0.800·9-s + 11-s − 7.55·12-s + 1.52·13-s + 6.06·14-s + 11.7·16-s + 3.39·17-s − 2.13·18-s + 19-s − 3.38·21-s + 2.66·22-s − 4.55·23-s − 12.2·24-s + 4.06·26-s + 5.63·27-s + 11.6·28-s + 9.48·29-s + 0.999·31-s + 14.8·32-s − 1.48·33-s + 9.03·34-s + ⋯ |
| L(s) = 1 | + 1.88·2-s − 0.856·3-s + 2.54·4-s − 1.61·6-s + 0.861·7-s + 2.91·8-s − 0.266·9-s + 0.301·11-s − 2.18·12-s + 0.423·13-s + 1.62·14-s + 2.93·16-s + 0.822·17-s − 0.502·18-s + 0.229·19-s − 0.737·21-s + 0.567·22-s − 0.950·23-s − 2.49·24-s + 0.797·26-s + 1.08·27-s + 2.19·28-s + 1.76·29-s + 0.179·31-s + 2.62·32-s − 0.258·33-s + 1.54·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.059500197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.059500197\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.66T + 2T^{2} \) |
| 3 | \( 1 + 1.48T + 3T^{2} \) |
| 7 | \( 1 - 2.27T + 7T^{2} \) |
| 13 | \( 1 - 1.52T + 13T^{2} \) |
| 17 | \( 1 - 3.39T + 17T^{2} \) |
| 23 | \( 1 + 4.55T + 23T^{2} \) |
| 29 | \( 1 - 9.48T + 29T^{2} \) |
| 31 | \( 1 - 0.999T + 31T^{2} \) |
| 37 | \( 1 + 7.35T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 - 6.26T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 9.11T + 61T^{2} \) |
| 67 | \( 1 - 1.97T + 67T^{2} \) |
| 71 | \( 1 + 9.85T + 71T^{2} \) |
| 73 | \( 1 - 5.69T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 9.93T + 83T^{2} \) |
| 89 | \( 1 + 3.48T + 89T^{2} \) |
| 97 | \( 1 - 6.80T + 97T^{2} \) |
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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
−8.041167121827753197147582384359, −7.02007384349443154686808407410, −6.47918317906122712375027813544, −5.81038398055001901880857478156, −5.20017131297909700396053495158, −4.76237501057299153918502532267, −3.86607310948426541451279745078, −3.15466091935556520279008783135, −2.13351766515476776485089600547, −1.13023772201400586268082681887,
1.13023772201400586268082681887, 2.13351766515476776485089600547, 3.15466091935556520279008783135, 3.86607310948426541451279745078, 4.76237501057299153918502532267, 5.20017131297909700396053495158, 5.81038398055001901880857478156, 6.47918317906122712375027813544, 7.02007384349443154686808407410, 8.041167121827753197147582384359
