| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 4·7-s + 8-s + 9-s − 4·11-s − 12-s − 4·13-s − 4·14-s + 16-s − 4·17-s + 18-s − 4·19-s + 4·21-s − 4·22-s − 8·23-s − 24-s − 4·26-s − 27-s − 4·28-s − 6·29-s − 4·31-s + 32-s + 4·33-s − 4·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s − 1.10·13-s − 1.06·14-s + 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.917·19-s + 0.872·21-s − 0.852·22-s − 1.66·23-s − 0.204·24-s − 0.784·26-s − 0.192·27-s − 0.755·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.696·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−12.97871825651029, −12.59133814410196, −12.23588706681045, −11.78172264349594, −11.03957449824477, −10.79922366717744, −10.31104748440659, −9.841311370288721, −9.444840047581023, −8.985365605475099, −8.170418017212826, −7.645174250498419, −7.321420206171411, −6.742159891643292, −6.230073762364179, −5.914583854547515, −5.491243353671746, −4.814295874564124, −4.353573739250485, −3.908801004797736, −3.287263017453425, −2.594809490900584, −2.307976632845564, −1.690605686090469, −0.3841177223441110, 0,
0.3841177223441110, 1.690605686090469, 2.307976632845564, 2.594809490900584, 3.287263017453425, 3.908801004797736, 4.353573739250485, 4.814295874564124, 5.491243353671746, 5.914583854547515, 6.230073762364179, 6.742159891643292, 7.321420206171411, 7.645174250498419, 8.170418017212826, 8.985365605475099, 9.444840047581023, 9.841311370288721, 10.31104748440659, 10.79922366717744, 11.03957449824477, 11.78172264349594, 12.23588706681045, 12.59133814410196, 12.97871825651029
