| L(s) = 1 | + 2.19·3-s + 5·5-s + 7·7-s − 22.1·9-s + 2.08·11-s + 86.2·13-s + 10.9·15-s − 39.3·17-s − 87.0·19-s + 15.3·21-s + 171.·23-s + 25·25-s − 107.·27-s − 28.4·29-s + 242.·31-s + 4.57·33-s + 35·35-s − 210.·37-s + 189.·39-s + 427.·41-s + 15.2·43-s − 110.·45-s − 496.·47-s + 49·49-s − 86.3·51-s + 169.·53-s + 10.4·55-s + ⋯ |
| L(s) = 1 | + 0.422·3-s + 0.447·5-s + 0.377·7-s − 0.821·9-s + 0.0571·11-s + 1.83·13-s + 0.188·15-s − 0.561·17-s − 1.05·19-s + 0.159·21-s + 1.55·23-s + 0.200·25-s − 0.769·27-s − 0.182·29-s + 1.40·31-s + 0.0241·33-s + 0.169·35-s − 0.935·37-s + 0.777·39-s + 1.62·41-s + 0.0539·43-s − 0.367·45-s − 1.54·47-s + 0.142·49-s − 0.237·51-s + 0.439·53-s + 0.0255·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.899355727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.899355727\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 - 7T \) |
| good | 3 | \( 1 - 2.19T + 27T^{2} \) |
| 11 | \( 1 - 2.08T + 1.33e3T^{2} \) |
| 13 | \( 1 - 86.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 87.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 171.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 28.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 242.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 210.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 427.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 15.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 496.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 169.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 369.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 655.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 404.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 50.3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 64.7T + 3.89e5T^{2} \) |
| 79 | \( 1 - 296.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 657.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 598.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 506.T + 9.12e5T^{2} \) |
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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
−9.180190370729932474240519234811, −8.632160101254962578632042280759, −8.142473382234585391922252116973, −6.79766790541351501410983057330, −6.14674765509760229001548418416, −5.22975450204384388817698815567, −4.12197514822342436299988837551, −3.11661594125639455503365270242, −2.09843601496089526855561913233, −0.893757332108311557769150745564,
0.893757332108311557769150745564, 2.09843601496089526855561913233, 3.11661594125639455503365270242, 4.12197514822342436299988837551, 5.22975450204384388817698815567, 6.14674765509760229001548418416, 6.79766790541351501410983057330, 8.142473382234585391922252116973, 8.632160101254962578632042280759, 9.180190370729932474240519234811
