| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s + 11-s + 2·13-s + 16-s + 17-s − 3·19-s − 2·20-s + 22-s − 23-s − 25-s + 2·26-s − 29-s − 2·31-s + 32-s + 34-s − 5·37-s − 3·38-s − 2·40-s − 10·41-s + 43-s + 44-s − 46-s + 7·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s + 0.301·11-s + 0.554·13-s + 1/4·16-s + 0.242·17-s − 0.688·19-s − 0.447·20-s + 0.213·22-s − 0.208·23-s − 1/5·25-s + 0.392·26-s − 0.185·29-s − 0.359·31-s + 0.176·32-s + 0.171·34-s − 0.821·37-s − 0.486·38-s − 0.316·40-s − 1.56·41-s + 0.152·43-s + 0.150·44-s − 0.147·46-s + 1.02·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.21595367049508057928776066431, −6.68780969747792011510707628397, −5.89548555527395535576149496329, −5.28027431718388085753161750342, −4.37640965511282924337532699376, −3.85882314640948234867650800558, −3.30893944263956688405362735139, −2.28631767824390458349990151964, −1.33984965538686538449952330658, 0,
1.33984965538686538449952330658, 2.28631767824390458349990151964, 3.30893944263956688405362735139, 3.85882314640948234867650800558, 4.37640965511282924337532699376, 5.28027431718388085753161750342, 5.89548555527395535576149496329, 6.68780969747792011510707628397, 7.21595367049508057928776066431
