| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s − 4·11-s + 12-s − 2·13-s − 15-s + 16-s + 18-s + 4·19-s − 20-s − 4·22-s + 24-s + 25-s − 2·26-s + 27-s + 2·29-s − 30-s − 8·31-s + 32-s − 4·33-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.258·15-s + 1/4·16-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.852·22-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.371·29-s − 0.182·30-s − 1.43·31-s + 0.176·32-s − 0.696·33-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−7.40561162349510633016588055418, −6.91954332593476671184463510324, −5.90592644132209508720459815133, −5.16366564361131865287337583164, −4.71364359659163093237904792022, −3.74770029001862206571218490220, −3.12938112745889251380126144765, −2.47906030853463410883102479535, −1.51008325733353209593744825657, 0,
1.51008325733353209593744825657, 2.47906030853463410883102479535, 3.12938112745889251380126144765, 3.74770029001862206571218490220, 4.71364359659163093237904792022, 5.16366564361131865287337583164, 5.90592644132209508720459815133, 6.91954332593476671184463510324, 7.40561162349510633016588055418
