| L(s) = 1 | − 495·2-s − 1.71e4·4-s − 3.35e6·5-s + 7.22e7·7-s + 1.38e8·8-s + 3.87e8·9-s + 1.66e9·10-s − 3.57e10·14-s − 6.39e10·16-s − 1.91e11·18-s − 3.01e11·19-s + 5.74e10·20-s + 7.44e12·25-s − 1.23e12·28-s − 2.64e13·31-s − 4.58e12·32-s − 2.42e14·35-s − 6.63e12·36-s + 1.49e14·38-s − 4.63e14·40-s + 6.40e14·41-s − 1.30e15·45-s − 4.84e14·47-s + 3.59e15·49-s − 3.68e15·50-s + 9.98e15·56-s − 1.47e16·59-s + ⋯ |
| L(s) = 1 | − 0.966·2-s − 0.0653·4-s − 1.71·5-s + 1.79·7-s + 1.02·8-s + 9-s + 1.66·10-s − 1.73·14-s − 0.930·16-s − 0.966·18-s − 0.934·19-s + 0.112·20-s + 1.95·25-s − 0.116·28-s − 31-s − 0.130·32-s − 3.07·35-s − 0.0653·36-s + 0.903·38-s − 1.76·40-s + 1.95·41-s − 1.71·45-s − 0.432·47-s + 2.20·49-s − 1.88·50-s + 1.84·56-s − 1.70·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(0.9325914305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9325914305\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + p^{9} T \) |
| good | 2 | \( 1 + 495 T + p^{18} T^{2} \) |
| 3 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 5 | \( 1 + 3355686 T + p^{18} T^{2} \) |
| 7 | \( 1 - 72260210 T + p^{18} T^{2} \) |
| 11 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 13 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 17 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 19 | \( 1 + 301505744342 T + p^{18} T^{2} \) |
| 23 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 29 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 37 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 41 | \( 1 - 640887818618322 T + p^{18} T^{2} \) |
| 43 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 47 | \( 1 + 484327216285470 T + p^{18} T^{2} \) |
| 53 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 59 | \( 1 + 14753739003245478 T + p^{18} T^{2} \) |
| 61 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 67 | \( 1 - 33886344531727690 T + p^{18} T^{2} \) |
| 71 | \( 1 + 86260446147898062 T + p^{18} T^{2} \) |
| 73 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 79 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 83 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 89 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 97 | \( 1 - 1388340453162394370 T + p^{18} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.67845650278808703327645762466, −11.35854022486146393672938933941, −10.63610399073561460039451573643, −8.885730664245830204405796122131, −7.910012607207070659030372306191, −7.38970743570324276010840878336, −4.69704901047632568819861724811, −4.09507367208330216107926038283, −1.71411407568811048568923496985, −0.63524853940931230639204006767,
0.63524853940931230639204006767, 1.71411407568811048568923496985, 4.09507367208330216107926038283, 4.69704901047632568819861724811, 7.38970743570324276010840878336, 7.910012607207070659030372306191, 8.885730664245830204405796122131, 10.63610399073561460039451573643, 11.35854022486146393672938933941, 12.67845650278808703327645762466
