L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 7-s − 3·8-s + 9-s − 11-s + 12-s + 2·13-s − 14-s − 16-s − 2·17-s + 18-s + 4·19-s + 21-s − 22-s + 3·24-s + 2·26-s − 27-s + 28-s + 6·29-s + 5·32-s + 33-s − 2·34-s − 36-s − 6·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.218·21-s − 0.213·22-s + 0.612·24-s + 0.392·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.883·32-s + 0.174·33-s − 0.342·34-s − 1/6·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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
−7.67929036297737109782771579041, −6.77652861978367889738682221111, −6.22954053136340011620718328316, −5.44149389611301332930143532743, −4.97039151884266236671711140089, −4.12948292404655658040477194414, −3.44202327772633783231803444075, −2.60512710788869817300582773740, −1.17364656672840219347653965769, 0,
1.17364656672840219347653965769, 2.60512710788869817300582773740, 3.44202327772633783231803444075, 4.12948292404655658040477194414, 4.97039151884266236671711140089, 5.44149389611301332930143532743, 6.22954053136340011620718328316, 6.77652861978367889738682221111, 7.67929036297737109782771579041