| L(s) = 1 | + 0.242·3-s + 0.379·5-s + 7-s − 2.94·9-s − 2.61·11-s − 2.31·13-s + 0.0919·15-s + 7.37·17-s − 5.44·19-s + 0.242·21-s + 6.44·23-s − 4.85·25-s − 1.43·27-s + 5.07·29-s + 6.10·31-s − 0.633·33-s + 0.379·35-s − 10.4·37-s − 0.560·39-s − 0.836·41-s + 5.50·43-s − 1.11·45-s − 6.02·47-s + 49-s + 1.78·51-s + 0.813·53-s − 0.992·55-s + ⋯ |
| L(s) = 1 | + 0.139·3-s + 0.169·5-s + 0.377·7-s − 0.980·9-s − 0.788·11-s − 0.641·13-s + 0.0237·15-s + 1.78·17-s − 1.24·19-s + 0.0528·21-s + 1.34·23-s − 0.971·25-s − 0.276·27-s + 0.941·29-s + 1.09·31-s − 0.110·33-s + 0.0641·35-s − 1.72·37-s − 0.0896·39-s − 0.130·41-s + 0.838·43-s − 0.166·45-s − 0.878·47-s + 0.142·49-s + 0.249·51-s + 0.111·53-s − 0.133·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 3 | \( 1 - 0.242T + 3T^{2} \) |
| 5 | \( 1 - 0.379T + 5T^{2} \) |
| 11 | \( 1 + 2.61T + 11T^{2} \) |
| 13 | \( 1 + 2.31T + 13T^{2} \) |
| 17 | \( 1 - 7.37T + 17T^{2} \) |
| 19 | \( 1 + 5.44T + 19T^{2} \) |
| 23 | \( 1 - 6.44T + 23T^{2} \) |
| 29 | \( 1 - 5.07T + 29T^{2} \) |
| 31 | \( 1 - 6.10T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 0.836T + 41T^{2} \) |
| 43 | \( 1 - 5.50T + 43T^{2} \) |
| 47 | \( 1 + 6.02T + 47T^{2} \) |
| 53 | \( 1 - 0.813T + 53T^{2} \) |
| 59 | \( 1 - 7.53T + 59T^{2} \) |
| 61 | \( 1 - 1.31T + 61T^{2} \) |
| 67 | \( 1 - 8.79T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 3.68T + 73T^{2} \) |
| 79 | \( 1 + 4.21T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 - 9.32T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.68522828381668310523418378774, −6.94879801367451349205663923933, −6.09091639879590567177830377132, −5.35081180679944895688757727024, −4.96903824786818195094012470461, −3.90856659982161983655468260835, −2.94947927851033178381136746650, −2.47184037466020622321820481914, −1.29723936560942170952227275283, 0,
1.29723936560942170952227275283, 2.47184037466020622321820481914, 2.94947927851033178381136746650, 3.90856659982161983655468260835, 4.96903824786818195094012470461, 5.35081180679944895688757727024, 6.09091639879590567177830377132, 6.94879801367451349205663923933, 7.68522828381668310523418378774
