L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 11-s − 2·13-s + 14-s + 16-s − 2·17-s − 4·19-s − 20-s + 22-s − 4·23-s + 25-s − 2·26-s + 28-s + 10·29-s − 4·31-s + 32-s − 2·34-s − 35-s − 10·37-s − 4·38-s − 40-s − 6·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.223·20-s + 0.213·22-s − 0.834·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s + 1.85·29-s − 0.718·31-s + 0.176·32-s − 0.342·34-s − 0.169·35-s − 1.64·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−7.45251335795537467807108002813, −6.85118163017637912182764577303, −6.22960856490657057114996084510, −5.36861235061873485534757182316, −4.61161020838167042656210265945, −4.14959404264402889674067304847, −3.26069483196783955185743536795, −2.37785815879164368956429366875, −1.51474658066086004336564780607, 0,
1.51474658066086004336564780607, 2.37785815879164368956429366875, 3.26069483196783955185743536795, 4.14959404264402889674067304847, 4.61161020838167042656210265945, 5.36861235061873485534757182316, 6.22960856490657057114996084510, 6.85118163017637912182764577303, 7.45251335795537467807108002813