L(s) = 1 | + 1.28·3-s − 4.29·5-s + 3.51·7-s − 1.34·9-s + 4.71·11-s − 3.44·13-s − 5.52·15-s − 1.72·17-s − 5.09·19-s + 4.52·21-s + 7.19·23-s + 13.4·25-s − 5.59·27-s − 4.72·29-s + 3.97·31-s + 6.07·33-s − 15.0·35-s − 8.45·37-s − 4.44·39-s + 1.87·41-s + 3.39·43-s + 5.75·45-s − 1.38·47-s + 5.34·49-s − 2.22·51-s + 1.09·53-s − 20.2·55-s + ⋯ |
L(s) = 1 | + 0.743·3-s − 1.91·5-s + 1.32·7-s − 0.446·9-s + 1.42·11-s − 0.956·13-s − 1.42·15-s − 0.419·17-s − 1.16·19-s + 0.987·21-s + 1.50·23-s + 2.68·25-s − 1.07·27-s − 0.877·29-s + 0.713·31-s + 1.05·33-s − 2.54·35-s − 1.39·37-s − 0.711·39-s + 0.292·41-s + 0.517·43-s + 0.857·45-s − 0.202·47-s + 0.763·49-s − 0.311·51-s + 0.150·53-s − 2.73·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 - T \) |
good | 3 | \( 1 - 1.28T + 3T^{2} \) |
| 5 | \( 1 + 4.29T + 5T^{2} \) |
| 7 | \( 1 - 3.51T + 7T^{2} \) |
| 11 | \( 1 - 4.71T + 11T^{2} \) |
| 13 | \( 1 + 3.44T + 13T^{2} \) |
| 17 | \( 1 + 1.72T + 17T^{2} \) |
| 19 | \( 1 + 5.09T + 19T^{2} \) |
| 23 | \( 1 - 7.19T + 23T^{2} \) |
| 29 | \( 1 + 4.72T + 29T^{2} \) |
| 31 | \( 1 - 3.97T + 31T^{2} \) |
| 37 | \( 1 + 8.45T + 37T^{2} \) |
| 41 | \( 1 - 1.87T + 41T^{2} \) |
| 43 | \( 1 - 3.39T + 43T^{2} \) |
| 47 | \( 1 + 1.38T + 47T^{2} \) |
| 53 | \( 1 - 1.09T + 53T^{2} \) |
| 59 | \( 1 + 14.6T + 59T^{2} \) |
| 61 | \( 1 + 3.40T + 61T^{2} \) |
| 67 | \( 1 - 9.01T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 7.95T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.990969058116828156701905067311, −7.11850724974645461120803040532, −6.70900936205781044310311751959, −5.30842025907239823591118150897, −4.55329053325840574258733726845, −4.08792165494758252291172210142, −3.34443099070534210296155691524, −2.44085937782819615585385625792, −1.33433715120694600023254292162, 0,
1.33433715120694600023254292162, 2.44085937782819615585385625792, 3.34443099070534210296155691524, 4.08792165494758252291172210142, 4.55329053325840574258733726845, 5.30842025907239823591118150897, 6.70900936205781044310311751959, 7.11850724974645461120803040532, 7.990969058116828156701905067311