| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s − 3·7-s + 9-s − 4·11-s + 2·12-s − 6·14-s − 4·16-s − 17-s + 2·18-s + 4·19-s − 3·21-s − 8·22-s − 6·23-s + 27-s − 6·28-s − 2·29-s − 7·31-s − 8·32-s − 4·33-s − 2·34-s + 2·36-s − 2·37-s + 8·38-s − 2·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s − 1.13·7-s + 1/3·9-s − 1.20·11-s + 0.577·12-s − 1.60·14-s − 16-s − 0.242·17-s + 0.471·18-s + 0.917·19-s − 0.654·21-s − 1.70·22-s − 1.25·23-s + 0.192·27-s − 1.13·28-s − 0.371·29-s − 1.25·31-s − 1.41·32-s − 0.696·33-s − 0.342·34-s + 1/3·36-s − 0.328·37-s + 1.29·38-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4127819387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4127819387\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
| 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
−13.05419218273467, −12.85677542819552, −12.07558889713285, −11.95621395060550, −11.27979776909175, −10.68101789504964, −10.19053083734260, −9.676465424547103, −9.418838621811374, −8.639847565866382, −8.306136793871627, −7.594477473399798, −7.115981616394875, −6.751413140105689, −6.082969456472018, −5.602948988036461, −5.287370376245394, −4.668098066230819, −4.038602166690547, −3.516022216368101, −3.263802223794909, −2.618499545517336, −2.204802045086366, −1.450008291992002, −0.1217914805696919,
0.1217914805696919, 1.450008291992002, 2.204802045086366, 2.618499545517336, 3.263802223794909, 3.516022216368101, 4.038602166690547, 4.668098066230819, 5.287370376245394, 5.602948988036461, 6.082969456472018, 6.751413140105689, 7.115981616394875, 7.594477473399798, 8.306136793871627, 8.639847565866382, 9.418838621811374, 9.676465424547103, 10.19053083734260, 10.68101789504964, 11.27979776909175, 11.95621395060550, 12.07558889713285, 12.85677542819552, 13.05419218273467
