| L(s) = 1 | + 19.8·3-s − 59.6·7-s + 153·9-s − 252·11-s − 119.·13-s + 689.·17-s − 220·19-s − 1.18e3·21-s − 2.43e3·23-s − 1.79e3·27-s + 6.93e3·29-s − 6.75e3·31-s − 5.01e3·33-s − 1.39e4·37-s − 2.37e3·39-s − 198·41-s + 417.·43-s − 1.05e4·47-s − 1.32e4·49-s + 1.37e4·51-s − 5.82e3·53-s − 4.37e3·57-s − 2.46e4·59-s − 5.69e3·61-s − 9.13e3·63-s − 4.36e4·67-s − 4.84e4·69-s + ⋯ |
| L(s) = 1 | + 1.27·3-s − 0.460·7-s + 0.629·9-s − 0.627·11-s − 0.195·13-s + 0.578·17-s − 0.139·19-s − 0.587·21-s − 0.959·23-s − 0.472·27-s + 1.53·29-s − 1.26·31-s − 0.801·33-s − 1.67·37-s − 0.250·39-s − 0.0183·41-s + 0.0344·43-s − 0.695·47-s − 0.787·49-s + 0.739·51-s − 0.284·53-s − 0.178·57-s − 0.922·59-s − 0.196·61-s − 0.289·63-s − 1.18·67-s − 1.22·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 19.8T + 243T^{2} \) |
| 7 | \( 1 + 59.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + 252T + 1.61e5T^{2} \) |
| 13 | \( 1 + 119.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 689.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 220T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.43e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.93e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.75e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.39e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 198T + 1.15e8T^{2} \) |
| 43 | \( 1 - 417.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.82e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.46e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.36e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.99e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.01e5T + 8.58e9T^{2} \) |
| 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
−9.883887973933525492609155786275, −9.024465740310844799764981421558, −8.183758468446020490020128732260, −7.47399857116392210165083014456, −6.28388660677356580970282665761, −5.03330167465079049744988522040, −3.67262407964577863461859941301, −2.89022068347581338570184162864, −1.79199034474100036482378086367, 0,
1.79199034474100036482378086367, 2.89022068347581338570184162864, 3.67262407964577863461859941301, 5.03330167465079049744988522040, 6.28388660677356580970282665761, 7.47399857116392210165083014456, 8.183758468446020490020128732260, 9.024465740310844799764981421558, 9.883887973933525492609155786275
