| L(s) = 1 | + 4.08·5-s − 7-s + 6.32·11-s + 3.09·13-s − 6.03·17-s − 0.726·19-s − 23-s + 11.6·25-s − 1.59·29-s + 5.45·31-s − 4.08·35-s − 5.46·37-s + 3.85·41-s + 5.67·43-s − 12.2·47-s + 49-s + 12.8·53-s + 25.7·55-s + 14.2·59-s + 11.8·61-s + 12.6·65-s − 1.47·67-s − 11.2·71-s − 8.08·73-s − 6.32·77-s + 7.14·79-s − 8.69·83-s + ⋯ |
| L(s) = 1 | + 1.82·5-s − 0.377·7-s + 1.90·11-s + 0.858·13-s − 1.46·17-s − 0.166·19-s − 0.208·23-s + 2.33·25-s − 0.295·29-s + 0.979·31-s − 0.689·35-s − 0.898·37-s + 0.601·41-s + 0.865·43-s − 1.79·47-s + 0.142·49-s + 1.76·53-s + 3.47·55-s + 1.85·59-s + 1.51·61-s + 1.56·65-s − 0.180·67-s − 1.33·71-s − 0.946·73-s − 0.720·77-s + 0.804·79-s − 0.954·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.319449074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.319449074\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 4.08T + 5T^{2} \) |
| 11 | \( 1 - 6.32T + 11T^{2} \) |
| 13 | \( 1 - 3.09T + 13T^{2} \) |
| 17 | \( 1 + 6.03T + 17T^{2} \) |
| 19 | \( 1 + 0.726T + 19T^{2} \) |
| 29 | \( 1 + 1.59T + 29T^{2} \) |
| 31 | \( 1 - 5.45T + 31T^{2} \) |
| 37 | \( 1 + 5.46T + 37T^{2} \) |
| 41 | \( 1 - 3.85T + 41T^{2} \) |
| 43 | \( 1 - 5.67T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 1.47T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + 8.08T + 73T^{2} \) |
| 79 | \( 1 - 7.14T + 79T^{2} \) |
| 83 | \( 1 + 8.69T + 83T^{2} \) |
| 89 | \( 1 - 3.50T + 89T^{2} \) |
| 97 | \( 1 - 4.57T + 97T^{2} \) |
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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
−8.510254644238796858491288317630, −6.94153923431450301416303843094, −6.63333125898496629173505263386, −6.09595554271843609601975976182, −5.45570868558999060277080481366, −4.40063786659121315642817813014, −3.73324942958480542781919034361, −2.59344112195983563916544641015, −1.83498831793838580608786633867, −1.04685979270425436744465471410,
1.04685979270425436744465471410, 1.83498831793838580608786633867, 2.59344112195983563916544641015, 3.73324942958480542781919034361, 4.40063786659121315642817813014, 5.45570868558999060277080481366, 6.09595554271843609601975976182, 6.63333125898496629173505263386, 6.94153923431450301416303843094, 8.510254644238796858491288317630
