L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s − 6·13-s + 14-s + 16-s + 6·19-s + 20-s − 22-s − 4·23-s + 25-s + 6·26-s − 28-s + 2·29-s + 4·31-s − 32-s − 35-s − 12·37-s − 6·38-s − 40-s + 2·41-s + 6·43-s + 44-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 − 1.66·13-s + 0.267·14-s + 1/4·16-s + 1.37·19-s + 0.223·20-s − 0.213·22-s − 0.834·23-s + 1/5·25-s + 1.17·26-s − 0.188·28-s + 0.371·29-s + 0.718·31-s − 0.176·32-s − 0.169·35-s − 1.97·37-s − 0.973·38-s − 0.158·40-s + 0.312·41-s + 0.914·43-s + 0.150·44-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 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−7.55460407730694378021877194786, −7.03855046615157838557552728009, −6.38485001647481648603295929390, −5.50111194514034854736292346912, −4.94549316134359719566249050259, −3.85960543735152724934769853802, −2.90894721432896775965863908181, −2.25328386941589949833792874145, −1.21585998214723947752686109049, 0,
1.21585998214723947752686109049, 2.25328386941589949833792874145, 2.90894721432896775965863908181, 3.85960543735152724934769853802, 4.94549316134359719566249050259, 5.50111194514034854736292346912, 6.38485001647481648603295929390, 7.03855046615157838557552728009, 7.55460407730694378021877194786