| L(s) = 1 | + 3·3-s + 4.51·5-s + 7.48·7-s + 9·9-s + 66.8·11-s − 13·13-s + 13.5·15-s + 96.9·17-s − 31.4·19-s + 22.4·21-s − 183.·23-s − 104.·25-s + 27·27-s + 112.·29-s + 77.2·31-s + 200.·33-s + 33.7·35-s + 54.7·37-s − 39·39-s + 451.·41-s + 113.·43-s + 40.6·45-s + 42.2·47-s − 287·49-s + 290.·51-s − 530.·53-s + 302.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.403·5-s + 0.404·7-s + 0.333·9-s + 1.83·11-s − 0.277·13-s + 0.233·15-s + 1.38·17-s − 0.380·19-s + 0.233·21-s − 1.66·23-s − 0.836·25-s + 0.192·27-s + 0.718·29-s + 0.447·31-s + 1.05·33-s + 0.163·35-s + 0.243·37-s − 0.160·39-s + 1.72·41-s + 0.402·43-s + 0.134·45-s + 0.131·47-s − 0.836·49-s + 0.798·51-s − 1.37·53-s + 0.740·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.227729120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.227729120\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 - 4.51T + 125T^{2} \) |
| 7 | \( 1 - 7.48T + 343T^{2} \) |
| 11 | \( 1 - 66.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 96.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 77.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 54.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 451.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 113.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 42.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 530.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 219.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 822.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 872.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 100.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 165.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 545.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 454.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 230.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.08e3T + 9.12e5T^{2} \) |
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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
−9.770701323103420460431909870279, −9.614037503897536650535820916523, −8.392720086460778074207726139323, −7.74692744715487557177130212898, −6.56774233062903416074053924048, −5.79946069360731658159358075157, −4.40826343846619671187500119753, −3.61143156369636993235568242995, −2.18899398287364693255483577102, −1.14245332056406843797872160453,
1.14245332056406843797872160453, 2.18899398287364693255483577102, 3.61143156369636993235568242995, 4.40826343846619671187500119753, 5.79946069360731658159358075157, 6.56774233062903416074053924048, 7.74692744715487557177130212898, 8.392720086460778074207726139323, 9.614037503897536650535820916523, 9.770701323103420460431909870279
