| L(s) = 1 | + 2.87·3-s + 1.81·5-s − 4.67·7-s + 5.25·9-s − 5.05·11-s − 2.90·13-s + 5.21·15-s + 4.51·17-s + 0.989·19-s − 13.4·21-s + 3.68·23-s − 1.70·25-s + 6.46·27-s + 2.64·29-s − 6.99·31-s − 14.5·33-s − 8.48·35-s + 7.54·37-s − 8.33·39-s − 4.10·41-s + 3.93·43-s + 9.53·45-s − 12.1·47-s + 14.8·49-s + 12.9·51-s − 12.7·53-s − 9.18·55-s + ⋯ |
| L(s) = 1 | + 1.65·3-s + 0.812·5-s − 1.76·7-s + 1.75·9-s − 1.52·11-s − 0.804·13-s + 1.34·15-s + 1.09·17-s + 0.227·19-s − 2.93·21-s + 0.769·23-s − 0.340·25-s + 1.24·27-s + 0.491·29-s − 1.25·31-s − 2.53·33-s − 1.43·35-s + 1.24·37-s − 1.33·39-s − 0.641·41-s + 0.600·43-s + 1.42·45-s − 1.76·47-s + 2.12·49-s + 1.81·51-s − 1.74·53-s − 1.23·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 2.87T + 3T^{2} \) |
| 5 | \( 1 - 1.81T + 5T^{2} \) |
| 7 | \( 1 + 4.67T + 7T^{2} \) |
| 11 | \( 1 + 5.05T + 11T^{2} \) |
| 13 | \( 1 + 2.90T + 13T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 19 | \( 1 - 0.989T + 19T^{2} \) |
| 23 | \( 1 - 3.68T + 23T^{2} \) |
| 29 | \( 1 - 2.64T + 29T^{2} \) |
| 31 | \( 1 + 6.99T + 31T^{2} \) |
| 37 | \( 1 - 7.54T + 37T^{2} \) |
| 41 | \( 1 + 4.10T + 41T^{2} \) |
| 43 | \( 1 - 3.93T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 4.97T + 59T^{2} \) |
| 61 | \( 1 + 6.41T + 61T^{2} \) |
| 67 | \( 1 + 4.76T + 67T^{2} \) |
| 71 | \( 1 + 3.33T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 4.74T + 97T^{2} \) |
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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
−7.58370548116681278656732534609, −7.02289390123096396847187598660, −6.11611684444888492887169431912, −5.47131451222663057330911243021, −4.57855154686134906569981542393, −3.43724728187933721883882652900, −2.97828580952603402144008509328, −2.59975520058844149603537482536, −1.59989237722748882436519627168, 0,
1.59989237722748882436519627168, 2.59975520058844149603537482536, 2.97828580952603402144008509328, 3.43724728187933721883882652900, 4.57855154686134906569981542393, 5.47131451222663057330911243021, 6.11611684444888492887169431912, 7.02289390123096396847187598660, 7.58370548116681278656732534609
