| L(s) = 1 | − 3.22·3-s + 1.94·5-s + 7-s + 7.38·9-s − 11-s − 13-s − 6.26·15-s + 4.63·17-s − 4.17·19-s − 3.22·21-s + 6.50·23-s − 1.21·25-s − 14.1·27-s + 3.35·29-s − 7.82·31-s + 3.22·33-s + 1.94·35-s + 3.35·37-s + 3.22·39-s − 5.66·41-s − 1.16·43-s + 14.3·45-s + 10.7·47-s + 49-s − 14.9·51-s − 11.5·53-s − 1.94·55-s + ⋯ |
| L(s) = 1 | − 1.86·3-s + 0.869·5-s + 0.377·7-s + 2.46·9-s − 0.301·11-s − 0.277·13-s − 1.61·15-s + 1.12·17-s − 0.958·19-s − 0.703·21-s + 1.35·23-s − 0.243·25-s − 2.71·27-s + 0.622·29-s − 1.40·31-s + 0.560·33-s + 0.328·35-s + 0.551·37-s + 0.515·39-s − 0.885·41-s − 0.178·43-s + 2.14·45-s + 1.56·47-s + 0.142·49-s − 2.09·51-s − 1.59·53-s − 0.262·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 3.22T + 3T^{2} \) |
| 5 | \( 1 - 1.94T + 5T^{2} \) |
| 17 | \( 1 - 4.63T + 17T^{2} \) |
| 19 | \( 1 + 4.17T + 19T^{2} \) |
| 23 | \( 1 - 6.50T + 23T^{2} \) |
| 29 | \( 1 - 3.35T + 29T^{2} \) |
| 31 | \( 1 + 7.82T + 31T^{2} \) |
| 37 | \( 1 - 3.35T + 37T^{2} \) |
| 41 | \( 1 + 5.66T + 41T^{2} \) |
| 43 | \( 1 + 1.16T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 4.96T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 0.157T + 71T^{2} \) |
| 73 | \( 1 - 1.45T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 4.26T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 6.52T + 97T^{2} \) |
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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
−7.27212830246378525609685255145, −6.64758829355381404149655249601, −5.87204420002162782252797797618, −5.58044286939944729772038404236, −4.86209125628582760737052866113, −4.31160115690270318015732451028, −3.08941218201288324075585218942, −1.86420606682495753654733450056, −1.18144728425797136419670867654, 0,
1.18144728425797136419670867654, 1.86420606682495753654733450056, 3.08941218201288324075585218942, 4.31160115690270318015732451028, 4.86209125628582760737052866113, 5.58044286939944729772038404236, 5.87204420002162782252797797618, 6.64758829355381404149655249601, 7.27212830246378525609685255145
