| L(s) = 1 | + 6·3-s − 8·5-s + 9·9-s − 56·11-s + 28·13-s − 48·15-s + 90·17-s + 74·19-s + 96·23-s − 61·25-s − 108·27-s − 222·29-s − 100·31-s − 336·33-s + 58·37-s + 168·39-s − 422·41-s − 512·43-s − 72·45-s + 148·47-s + 540·51-s − 642·53-s + 448·55-s + 444·57-s − 318·59-s − 720·61-s − 224·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.715·5-s + 1/3·9-s − 1.53·11-s + 0.597·13-s − 0.826·15-s + 1.28·17-s + 0.893·19-s + 0.870·23-s − 0.487·25-s − 0.769·27-s − 1.42·29-s − 0.579·31-s − 1.77·33-s + 0.257·37-s + 0.689·39-s − 1.60·41-s − 1.81·43-s − 0.238·45-s + 0.459·47-s + 1.48·51-s − 1.66·53-s + 1.09·55-s + 1.03·57-s − 0.701·59-s − 1.51·61-s − 0.427·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 5 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 56 T + p^{3} T^{2} \) |
| 13 | \( 1 - 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 90 T + p^{3} T^{2} \) |
| 19 | \( 1 - 74 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 + 222 T + p^{3} T^{2} \) |
| 31 | \( 1 + 100 T + p^{3} T^{2} \) |
| 37 | \( 1 - 58 T + p^{3} T^{2} \) |
| 41 | \( 1 + 422 T + p^{3} T^{2} \) |
| 43 | \( 1 + 512 T + p^{3} T^{2} \) |
| 47 | \( 1 - 148 T + p^{3} T^{2} \) |
| 53 | \( 1 + 642 T + p^{3} T^{2} \) |
| 59 | \( 1 + 318 T + p^{3} T^{2} \) |
| 61 | \( 1 + 720 T + p^{3} T^{2} \) |
| 67 | \( 1 - 412 T + p^{3} T^{2} \) |
| 71 | \( 1 + 448 T + p^{3} T^{2} \) |
| 73 | \( 1 + 994 T + p^{3} T^{2} \) |
| 79 | \( 1 - 296 T + p^{3} T^{2} \) |
| 83 | \( 1 - 386 T + p^{3} T^{2} \) |
| 89 | \( 1 - 6 T + p^{3} T^{2} \) |
| 97 | \( 1 - 138 T + p^{3} T^{2} \) |
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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
−9.379280704228272106499040754305, −8.473115963865702257903450793502, −7.76453072727515427067824410587, −7.40109030284861566637131544576, −5.80224917138952816022061552195, −4.93066755587831069603466742386, −3.42279428186260198787099823150, −3.17434245525254923117420114742, −1.71537766719769320310941150989, 0,
1.71537766719769320310941150989, 3.17434245525254923117420114742, 3.42279428186260198787099823150, 4.93066755587831069603466742386, 5.80224917138952816022061552195, 7.40109030284861566637131544576, 7.76453072727515427067824410587, 8.473115963865702257903450793502, 9.379280704228272106499040754305
