| L(s) = 1 | + 2.29·3-s − 1.25·7-s + 2.25·9-s − 2·11-s − 13-s − 4.80·17-s + 5.09·19-s − 2.87·21-s − 2.58·23-s − 1.70·27-s + 5.09·29-s − 8.58·31-s − 4.58·33-s + 7.83·37-s − 2.29·39-s − 9.67·41-s − 10.8·43-s − 2.74·47-s − 5.42·49-s − 11.0·51-s + 2.58·53-s + 11.6·57-s + 5.09·59-s + 13.6·61-s − 2.83·63-s − 8.58·67-s − 5.92·69-s + ⋯ |
| L(s) = 1 | + 1.32·3-s − 0.474·7-s + 0.751·9-s − 0.603·11-s − 0.277·13-s − 1.16·17-s + 1.16·19-s − 0.627·21-s − 0.538·23-s − 0.328·27-s + 0.946·29-s − 1.54·31-s − 0.798·33-s + 1.28·37-s − 0.367·39-s − 1.51·41-s − 1.65·43-s − 0.400·47-s − 0.774·49-s − 1.54·51-s + 0.355·53-s + 1.54·57-s + 0.663·59-s + 1.75·61-s − 0.356·63-s − 1.04·67-s − 0.713·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.29T + 3T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 + 4.80T + 17T^{2} \) |
| 19 | \( 1 - 5.09T + 19T^{2} \) |
| 23 | \( 1 + 2.58T + 23T^{2} \) |
| 29 | \( 1 - 5.09T + 29T^{2} \) |
| 31 | \( 1 + 8.58T + 31T^{2} \) |
| 37 | \( 1 - 7.83T + 37T^{2} \) |
| 41 | \( 1 + 9.67T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 2.74T + 47T^{2} \) |
| 53 | \( 1 - 2.58T + 53T^{2} \) |
| 59 | \( 1 - 5.09T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 + 8.58T + 67T^{2} \) |
| 71 | \( 1 + 5.38T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 5.09T + 89T^{2} \) |
| 97 | \( 1 + 6.26T + 97T^{2} \) |
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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
−7.996555427336631054946376539916, −7.23154556242278729986154771903, −6.64650799159870964544430085993, −5.62254382876655270606153151594, −4.83930329464010162110887780495, −3.88980056014297353346507698724, −3.16681091049106299102329269343, −2.54255453398850680524314247071, −1.66775719015641023432530164089, 0,
1.66775719015641023432530164089, 2.54255453398850680524314247071, 3.16681091049106299102329269343, 3.88980056014297353346507698724, 4.83930329464010162110887780495, 5.62254382876655270606153151594, 6.64650799159870964544430085993, 7.23154556242278729986154771903, 7.996555427336631054946376539916
