L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 4·7-s − 8-s + 9-s − 2·11-s − 12-s − 2·13-s + 4·14-s + 16-s + 2·17-s − 18-s + 2·19-s + 4·21-s + 2·22-s + 24-s + 2·26-s − 27-s − 4·28-s + 2·29-s + 4·31-s − 32-s + 2·33-s − 2·34-s + 36-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 − 0.603·11-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 + 0.458·19-s + 0.872·21-s + 0.426·22-s + 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.755·28-s + 0.371·29-s + 0.718·31-s − 0.176·32-s + 0.348·33-s − 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.74593670654687023683873447922, −7.00566116256809770507293385520, −6.54529828892243655106166867615, −5.73308229592129608621771128857, −5.12743183833987971514073272881, −3.97758013969812929029850281589, −3.08795703472397618089672444704, −2.38791136708214022248822531236, −0.972238064280248028855185164022, 0,
0.972238064280248028855185164022, 2.38791136708214022248822531236, 3.08795703472397618089672444704, 3.97758013969812929029850281589, 5.12743183833987971514073272881, 5.73308229592129608621771128857, 6.54529828892243655106166867615, 7.00566116256809770507293385520, 7.74593670654687023683873447922