| L(s) = 1 | − 4.39·2-s − 1.00e3·4-s − 5.97e3·5-s − 2.76e4·7-s + 8.92e3·8-s + 5.90e4·9-s + 2.62e4·10-s + 1.21e5·14-s + 9.89e5·16-s − 2.59e5·18-s − 4.92e6·19-s + 6.00e6·20-s + 2.59e7·25-s + 2.77e7·28-s − 2.86e7·31-s − 1.34e7·32-s + 1.65e8·35-s − 5.93e7·36-s + 2.16e7·38-s − 5.32e7·40-s + 1.59e8·41-s − 3.52e8·45-s − 4.58e8·47-s + 4.81e8·49-s − 1.13e8·50-s − 2.46e8·56-s − 6.61e8·59-s + ⋯ |
| L(s) = 1 | − 0.137·2-s − 0.981·4-s − 1.91·5-s − 1.64·7-s + 0.272·8-s + 0.999·9-s + 0.262·10-s + 0.225·14-s + 0.943·16-s − 0.137·18-s − 1.99·19-s + 1.87·20-s + 2.65·25-s + 1.61·28-s − 31-s − 0.401·32-s + 3.14·35-s − 0.981·36-s + 0.273·38-s − 0.520·40-s + 1.37·41-s − 1.91·45-s − 1.99·47-s + 1.70·49-s − 0.364·50-s − 0.447·56-s − 0.925·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.3460010487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3460010487\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + 2.86e7T \) |
| good | 2 | \( 1 + 4.39T + 1.02e3T^{2} \) |
| 3 | \( 1 - 5.90e4T^{2} \) |
| 5 | \( 1 + 5.97e3T + 9.76e6T^{2} \) |
| 7 | \( 1 + 2.76e4T + 2.82e8T^{2} \) |
| 11 | \( 1 - 2.59e10T^{2} \) |
| 13 | \( 1 - 1.37e11T^{2} \) |
| 17 | \( 1 - 2.01e12T^{2} \) |
| 19 | \( 1 + 4.92e6T + 6.13e12T^{2} \) |
| 23 | \( 1 - 4.14e13T^{2} \) |
| 29 | \( 1 - 4.20e14T^{2} \) |
| 37 | \( 1 - 4.80e15T^{2} \) |
| 41 | \( 1 - 1.59e8T + 1.34e16T^{2} \) |
| 43 | \( 1 - 2.16e16T^{2} \) |
| 47 | \( 1 + 4.58e8T + 5.25e16T^{2} \) |
| 53 | \( 1 - 1.74e17T^{2} \) |
| 59 | \( 1 + 6.61e8T + 5.11e17T^{2} \) |
| 61 | \( 1 - 7.13e17T^{2} \) |
| 67 | \( 1 - 9.85e8T + 1.82e18T^{2} \) |
| 71 | \( 1 - 3.56e9T + 3.25e18T^{2} \) |
| 73 | \( 1 - 4.29e18T^{2} \) |
| 79 | \( 1 - 9.46e18T^{2} \) |
| 83 | \( 1 - 1.55e19T^{2} \) |
| 89 | \( 1 - 3.11e19T^{2} \) |
| 97 | \( 1 + 4.91e9T + 7.37e19T^{2} \) |
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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
−14.86258433758893395469783173313, −12.91344303764927801653278479571, −12.55872701993247415478707665543, −10.71866738786433289532487616253, −9.371314451374981182527907977981, −8.084473889483491311956901454471, −6.76379737319788076589752417789, −4.37041780287634385452230741787, −3.55791117652996673590737127161, −0.40322986911043983562946478197,
0.40322986911043983562946478197, 3.55791117652996673590737127161, 4.37041780287634385452230741787, 6.76379737319788076589752417789, 8.084473889483491311956901454471, 9.371314451374981182527907977981, 10.71866738786433289532487616253, 12.55872701993247415478707665543, 12.91344303764927801653278479571, 14.86258433758893395469783173313
