| L(s) = 1 | + 1.50e4·2-s + 9.37e7·4-s + 1.22e9·5-s + 2.87e11·7-s − 6.11e11·8-s + 1.84e13·10-s + 1.77e14·11-s − 5.40e14·13-s + 4.33e15·14-s − 2.18e16·16-s − 5.58e15·17-s + 1.87e17·19-s + 1.14e17·20-s + 2.67e18·22-s + 1.25e18·23-s + 1.49e18·25-s − 8.15e18·26-s + 2.69e19·28-s − 5.09e19·29-s + 2.57e20·31-s − 2.47e20·32-s − 8.43e19·34-s + 3.50e20·35-s − 6.56e20·37-s + 2.83e21·38-s − 7.46e20·40-s − 4.64e20·41-s + ⋯ |
| L(s) = 1 | + 1.30·2-s + 0.698·4-s + 0.447·5-s + 1.12·7-s − 0.393·8-s + 0.582·10-s + 1.54·11-s − 0.494·13-s + 1.46·14-s − 1.21·16-s − 0.136·17-s + 1.02·19-s + 0.312·20-s + 2.01·22-s + 0.518·23-s + 0.199·25-s − 0.644·26-s + 0.782·28-s − 0.921·29-s + 1.89·31-s − 1.18·32-s − 0.178·34-s + 0.501·35-s − 0.442·37-s + 1.33·38-s − 0.175·40-s − 0.0784·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(14)\) |
\(\approx\) |
\(6.784194856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.784194856\) |
| \(L(\frac{29}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 1.22e9T \) |
| good | 2 | \( 1 - 1.50e4T + 1.34e8T^{2} \) |
| 7 | \( 1 - 2.87e11T + 6.57e22T^{2} \) |
| 11 | \( 1 - 1.77e14T + 1.31e28T^{2} \) |
| 13 | \( 1 + 5.40e14T + 1.19e30T^{2} \) |
| 17 | \( 1 + 5.58e15T + 1.66e33T^{2} \) |
| 19 | \( 1 - 1.87e17T + 3.36e34T^{2} \) |
| 23 | \( 1 - 1.25e18T + 5.84e36T^{2} \) |
| 29 | \( 1 + 5.09e19T + 3.05e39T^{2} \) |
| 31 | \( 1 - 2.57e20T + 1.84e40T^{2} \) |
| 37 | \( 1 + 6.56e20T + 2.19e42T^{2} \) |
| 41 | \( 1 + 4.64e20T + 3.50e43T^{2} \) |
| 43 | \( 1 - 7.04e21T + 1.26e44T^{2} \) |
| 47 | \( 1 - 7.81e21T + 1.40e45T^{2} \) |
| 53 | \( 1 + 3.58e23T + 3.59e46T^{2} \) |
| 59 | \( 1 - 1.07e24T + 6.50e47T^{2} \) |
| 61 | \( 1 - 2.22e24T + 1.59e48T^{2} \) |
| 67 | \( 1 - 1.96e23T + 2.01e49T^{2} \) |
| 71 | \( 1 + 1.57e25T + 9.63e49T^{2} \) |
| 73 | \( 1 - 5.36e24T + 2.04e50T^{2} \) |
| 79 | \( 1 + 5.98e25T + 1.72e51T^{2} \) |
| 83 | \( 1 + 1.26e25T + 6.53e51T^{2} \) |
| 89 | \( 1 + 4.25e25T + 4.30e52T^{2} \) |
| 97 | \( 1 - 1.03e27T + 4.39e53T^{2} \) |
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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
−11.38110450004971763246607053660, −9.711956843457976923895832847832, −8.668046910911374625124670414823, −7.15763451839537382614050349602, −6.07856913855654507680313147963, −5.05395104604014272062392582664, −4.31132632362974405834347521760, −3.18718146280525662266921440599, −1.97174239774110083910078432513, −0.937158493930197558558134677844,
0.937158493930197558558134677844, 1.97174239774110083910078432513, 3.18718146280525662266921440599, 4.31132632362974405834347521760, 5.05395104604014272062392582664, 6.07856913855654507680313147963, 7.15763451839537382614050349602, 8.668046910911374625124670414823, 9.711956843457976923895832847832, 11.38110450004971763246607053660
