L(s) = 1 | + 2-s + 3-s − 4-s + 6-s + 4·7-s − 3·8-s + 9-s − 12-s − 2·13-s + 4·14-s − 16-s − 2·17-s + 18-s + 4·21-s − 8·23-s − 3·24-s − 2·26-s + 27-s − 4·28-s + 6·29-s − 8·31-s + 5·32-s − 2·34-s − 36-s − 6·37-s − 2·39-s + 2·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s + 1.06·14-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.872·21-s − 1.66·23-s − 0.612·24-s − 0.392·26-s + 0.192·27-s − 0.755·28-s + 1.11·29-s − 1.43·31-s + 0.883·32-s − 0.342·34-s − 1/6·36-s − 0.986·37-s − 0.320·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.56703394334927679521241514429, −6.64989431233589188309119242243, −5.81886729913280632057895570273, −5.08451211650611410303030375999, −4.58601163808162548244809882989, −4.01898082351391503616891777906, −3.19345176929607987672594852845, −2.23718821890583247191367747104, −1.52741118326423034153490846090, 0,
1.52741118326423034153490846090, 2.23718821890583247191367747104, 3.19345176929607987672594852845, 4.01898082351391503616891777906, 4.58601163808162548244809882989, 5.08451211650611410303030375999, 5.81886729913280632057895570273, 6.64989431233589188309119242243, 7.56703394334927679521241514429