| L(s) = 1 | + 2·2-s + 4·4-s − 8·5-s + 8·8-s − 16·10-s − 40·11-s + 4·13-s + 16·16-s + 84·17-s + 148·19-s − 32·20-s − 80·22-s − 84·23-s − 61·25-s + 8·26-s − 58·29-s − 136·31-s + 32·32-s + 168·34-s − 222·37-s + 296·38-s − 64·40-s − 420·41-s − 164·43-s − 160·44-s − 168·46-s − 488·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.715·5-s + 0.353·8-s − 0.505·10-s − 1.09·11-s + 0.0853·13-s + 1/4·16-s + 1.19·17-s + 1.78·19-s − 0.357·20-s − 0.775·22-s − 0.761·23-s − 0.487·25-s + 0.0603·26-s − 0.371·29-s − 0.787·31-s + 0.176·32-s + 0.847·34-s − 0.986·37-s + 1.26·38-s − 0.252·40-s − 1.59·41-s − 0.581·43-s − 0.548·44-s − 0.538·46-s − 1.51·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 13 | \( 1 - 4 T + p^{3} T^{2} \) |
| 17 | \( 1 - 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 148 T + p^{3} T^{2} \) |
| 23 | \( 1 + 84 T + p^{3} T^{2} \) |
| 29 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 136 T + p^{3} T^{2} \) |
| 37 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 420 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 488 T + p^{3} T^{2} \) |
| 53 | \( 1 + 478 T + p^{3} T^{2} \) |
| 59 | \( 1 + 548 T + p^{3} T^{2} \) |
| 61 | \( 1 - 692 T + p^{3} T^{2} \) |
| 67 | \( 1 + 908 T + p^{3} T^{2} \) |
| 71 | \( 1 - 524 T + p^{3} T^{2} \) |
| 73 | \( 1 - 440 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1216 T + p^{3} T^{2} \) |
| 83 | \( 1 - 684 T + p^{3} T^{2} \) |
| 89 | \( 1 + 604 T + p^{3} T^{2} \) |
| 97 | \( 1 + 832 T + p^{3} T^{2} \) |
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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
−9.518891825966928071161096521297, −8.002021173179334595337055604577, −7.79724119885896491701904489856, −6.74988468406563153460018691238, −5.49591301424265376754342464628, −5.06117600581035053885485637600, −3.67334448281120564004341133587, −3.13297950197622409963040447226, −1.63448546339280333625053987790, 0,
1.63448546339280333625053987790, 3.13297950197622409963040447226, 3.67334448281120564004341133587, 5.06117600581035053885485637600, 5.49591301424265376754342464628, 6.74988468406563153460018691238, 7.79724119885896491701904489856, 8.002021173179334595337055604577, 9.518891825966928071161096521297
