| L(s) = 1 | + 4.82e5·3-s + 1.37e8·5-s + 1.92e9·7-s + 1.38e11·9-s + 1.11e10·11-s + 4.97e12·13-s + 6.60e13·15-s + 1.59e14·17-s + 6.91e14·19-s + 9.29e14·21-s − 7.11e15·23-s + 6.85e15·25-s + 2.12e16·27-s + 1.00e17·29-s − 3.03e16·31-s + 5.39e15·33-s + 2.64e17·35-s + 1.55e18·37-s + 2.39e18·39-s − 5.17e18·41-s − 6.33e18·43-s + 1.89e19·45-s − 8.39e18·47-s − 2.36e19·49-s + 7.69e19·51-s + 9.68e19·53-s + 1.53e18·55-s + ⋯ |
| L(s) = 1 | + 1.57·3-s + 1.25·5-s + 0.368·7-s + 1.46·9-s + 0.0118·11-s + 0.770·13-s + 1.97·15-s + 1.12·17-s + 1.36·19-s + 0.579·21-s − 1.55·23-s + 0.575·25-s + 0.737·27-s + 1.53·29-s − 0.214·31-s + 0.0185·33-s + 0.462·35-s + 1.43·37-s + 1.21·39-s − 1.46·41-s − 1.04·43-s + 1.84·45-s − 0.495·47-s − 0.864·49-s + 1.77·51-s + 1.43·53-s + 0.0148·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(7.221730151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.221730151\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 4.82e5T + 9.41e10T^{2} \) |
| 5 | \( 1 - 1.37e8T + 1.19e16T^{2} \) |
| 7 | \( 1 - 1.92e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 1.11e10T + 8.95e23T^{2} \) |
| 13 | \( 1 - 4.97e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 1.59e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 6.91e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 7.11e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 1.00e17T + 4.31e33T^{2} \) |
| 31 | \( 1 + 3.03e16T + 2.00e34T^{2} \) |
| 37 | \( 1 - 1.55e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + 5.17e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 6.33e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 8.39e18T + 2.87e38T^{2} \) |
| 53 | \( 1 - 9.68e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 3.08e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 1.66e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 1.26e21T + 9.99e41T^{2} \) |
| 71 | \( 1 - 3.09e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 2.48e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 1.08e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 1.23e22T + 1.37e44T^{2} \) |
| 89 | \( 1 + 4.40e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 9.25e22T + 4.96e45T^{2} \) |
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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
−10.08045542513286103271497249843, −9.668365922964731374492772430693, −8.479778088437787536878363684059, −7.78775869861601724527011617395, −6.34598214892251804086751895920, −5.20971042291690936436034283791, −3.73231754322165101478142970356, −2.84846846633108071764873856583, −1.85844500902842682385253107886, −1.14114677635969984072751653611,
1.14114677635969984072751653611, 1.85844500902842682385253107886, 2.84846846633108071764873856583, 3.73231754322165101478142970356, 5.20971042291690936436034283791, 6.34598214892251804086751895920, 7.78775869861601724527011617395, 8.479778088437787536878363684059, 9.668365922964731374492772430693, 10.08045542513286103271497249843
