| L(s) = 1 | + 7.82·3-s + 9.23·5-s + 34.1·9-s + 71.3·11-s + 46.3·13-s + 72.2·15-s + 83.7·17-s + 2.16·19-s − 193.·23-s − 39.6·25-s + 56.2·27-s − 135.·29-s − 20.5·31-s + 557.·33-s − 324.·37-s + 362.·39-s − 431.·41-s + 135.·43-s + 315.·45-s + 592.·47-s + 655.·51-s + 182.·53-s + 658.·55-s + 16.9·57-s − 208.·59-s − 80.2·61-s + 428.·65-s + ⋯ |
| L(s) = 1 | + 1.50·3-s + 0.826·5-s + 1.26·9-s + 1.95·11-s + 0.989·13-s + 1.24·15-s + 1.19·17-s + 0.0261·19-s − 1.75·23-s − 0.317·25-s + 0.400·27-s − 0.868·29-s − 0.119·31-s + 2.94·33-s − 1.44·37-s + 1.48·39-s − 1.64·41-s + 0.481·43-s + 1.04·45-s + 1.83·47-s + 1.79·51-s + 0.473·53-s + 1.61·55-s + 0.0393·57-s − 0.460·59-s − 0.168·61-s + 0.817·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.885356354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.885356354\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 7.82T + 27T^{2} \) |
| 5 | \( 1 - 9.23T + 125T^{2} \) |
| 11 | \( 1 - 71.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 46.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 83.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 2.16T + 6.85e3T^{2} \) |
| 23 | \( 1 + 193.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 135.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 20.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 324.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 431.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 135.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 592.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 182.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 208.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 80.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 831.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 59.1T + 3.57e5T^{2} \) |
| 73 | \( 1 + 367.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 438.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 35.0T + 5.71e5T^{2} \) |
| 89 | \( 1 - 824.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.29e3T + 9.12e5T^{2} \) |
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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
−9.653589356065668237022998757104, −9.056350178443039462447043816193, −8.412082241416079660111943441957, −7.47639748664951709187794590251, −6.42412349980696026918122951545, −5.62681695641465331378168337260, −3.86257811700947672546258638045, −3.59725436387820333009002516963, −2.07402296449894443350725400131, −1.38251344770435356644883585614,
1.38251344770435356644883585614, 2.07402296449894443350725400131, 3.59725436387820333009002516963, 3.86257811700947672546258638045, 5.62681695641465331378168337260, 6.42412349980696026918122951545, 7.47639748664951709187794590251, 8.412082241416079660111943441957, 9.056350178443039462447043816193, 9.653589356065668237022998757104
