| L(s) = 1 | + 2-s − 3·3-s + 4-s + 2·5-s − 3·6-s + 8-s + 6·9-s + 2·10-s + 2·11-s − 3·12-s − 13-s − 6·15-s + 16-s − 3·17-s + 6·18-s − 19-s + 2·20-s + 2·22-s + 23-s − 3·24-s − 25-s − 26-s − 9·27-s − 9·29-s − 6·30-s + 32-s − 6·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.894·5-s − 1.22·6-s + 0.353·8-s + 2·9-s + 0.632·10-s + 0.603·11-s − 0.866·12-s − 0.277·13-s − 1.54·15-s + 1/4·16-s − 0.727·17-s + 1.41·18-s − 0.229·19-s + 0.447·20-s + 0.426·22-s + 0.208·23-s − 0.612·24-s − 1/5·25-s − 0.196·26-s − 1.73·27-s − 1.67·29-s − 1.09·30-s + 0.176·32-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5194 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5194 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.37944278354610470124363514839, −6.77452635732619944656678916701, −6.29156692732698318909645788646, −5.58640252079145470815658631128, −5.16401203537076209655477088937, −4.38176664774152857343531744367, −3.57311651605977195970668961213, −2.14997408348872754556016192239, −1.44188190342239797850774335519, 0,
1.44188190342239797850774335519, 2.14997408348872754556016192239, 3.57311651605977195970668961213, 4.38176664774152857343531744367, 5.16401203537076209655477088937, 5.58640252079145470815658631128, 6.29156692732698318909645788646, 6.77452635732619944656678916701, 7.37944278354610470124363514839
