| L(s) = 1 | + 4·2-s − 16·4-s + 19·5-s + 10·7-s − 192·8-s + 76·10-s + 121·11-s − 1.14e3·13-s + 40·14-s − 256·16-s − 686·17-s − 384·19-s − 304·20-s + 484·22-s − 3.70e3·23-s − 2.76e3·25-s − 4.59e3·26-s − 160·28-s + 5.42e3·29-s − 6.44e3·31-s + 5.12e3·32-s − 2.74e3·34-s + 190·35-s + 1.20e4·37-s − 1.53e3·38-s − 3.64e3·40-s + 1.52e3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.339·5-s + 0.0771·7-s − 1.06·8-s + 0.240·10-s + 0.301·11-s − 1.88·13-s + 0.0545·14-s − 1/4·16-s − 0.575·17-s − 0.244·19-s − 0.169·20-s + 0.213·22-s − 1.46·23-s − 0.884·25-s − 1.33·26-s − 0.0385·28-s + 1.19·29-s − 1.20·31-s + 0.883·32-s − 0.407·34-s + 0.0262·35-s + 1.44·37-s − 0.172·38-s − 0.360·40-s + 0.141·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 - p^{2} T + p^{5} T^{2} \) |
| 5 | \( 1 - 19 T + p^{5} T^{2} \) |
| 7 | \( 1 - 10 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1148 T + p^{5} T^{2} \) |
| 17 | \( 1 + 686 T + p^{5} T^{2} \) |
| 19 | \( 1 + 384 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3709 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5424 T + p^{5} T^{2} \) |
| 31 | \( 1 + 6443 T + p^{5} T^{2} \) |
| 37 | \( 1 - 12063 T + p^{5} T^{2} \) |
| 41 | \( 1 - 1528 T + p^{5} T^{2} \) |
| 43 | \( 1 + 4026 T + p^{5} T^{2} \) |
| 47 | \( 1 + 7168 T + p^{5} T^{2} \) |
| 53 | \( 1 - 29862 T + p^{5} T^{2} \) |
| 59 | \( 1 - 6461 T + p^{5} T^{2} \) |
| 61 | \( 1 + 16980 T + p^{5} T^{2} \) |
| 67 | \( 1 - 29999 T + p^{5} T^{2} \) |
| 71 | \( 1 + 31023 T + p^{5} T^{2} \) |
| 73 | \( 1 - 1924 T + p^{5} T^{2} \) |
| 79 | \( 1 - 65138 T + p^{5} T^{2} \) |
| 83 | \( 1 - 102714 T + p^{5} T^{2} \) |
| 89 | \( 1 + 17415 T + p^{5} T^{2} \) |
| 97 | \( 1 - 66905 T + p^{5} T^{2} \) |
| 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
−12.50950859468504575738360007063, −11.73366747322782986789084884997, −10.07118556912476989321768739592, −9.292976145996946453589101210740, −7.904305247890995823725828211671, −6.38415385656017770797677190105, −5.14991144505655768985694924122, −4.08941014976183756182469263837, −2.37779347924315653506074882625, 0,
2.37779347924315653506074882625, 4.08941014976183756182469263837, 5.14991144505655768985694924122, 6.38415385656017770797677190105, 7.904305247890995823725828211671, 9.292976145996946453589101210740, 10.07118556912476989321768739592, 11.73366747322782986789084884997, 12.50950859468504575738360007063
