| L(s) = 1 | + 3-s − 5-s + 9-s − 4·11-s − 2·13-s − 15-s − 2·17-s + 19-s + 8·23-s + 25-s + 27-s + 6·29-s − 8·31-s − 4·33-s − 10·37-s − 2·39-s − 2·41-s − 8·43-s − 45-s − 8·47-s − 7·49-s − 2·51-s − 6·53-s + 4·55-s + 57-s − 8·59-s − 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s − 0.485·17-s + 0.229·19-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s − 1.64·37-s − 0.320·39-s − 0.312·41-s − 1.21·43-s − 0.149·45-s − 1.16·47-s − 49-s − 0.280·51-s − 0.824·53-s + 0.539·55-s + 0.132·57-s − 1.04·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−8.596701273298697380060824995911, −7.88238895909905781604956925371, −7.21640628490768978119201784847, −6.51317057482729735885667090707, −5.07764951701936065593166424954, −4.84901790522410675786139648038, −3.44471499214125281526670115019, −2.88541236131145081348127915003, −1.71156881481067557903848642603, 0,
1.71156881481067557903848642603, 2.88541236131145081348127915003, 3.44471499214125281526670115019, 4.84901790522410675786139648038, 5.07764951701936065593166424954, 6.51317057482729735885667090707, 7.21640628490768978119201784847, 7.88238895909905781604956925371, 8.596701273298697380060824995911
