| L(s) = 1 | + 2·3-s + 2·7-s + 9-s + 13-s + 5·17-s − 6·19-s + 4·21-s − 2·23-s − 4·27-s − 9·29-s − 2·31-s − 3·37-s + 2·39-s − 5·41-s − 2·47-s − 3·49-s + 10·51-s + 9·53-s − 12·57-s − 8·59-s − 6·61-s + 2·63-s + 2·67-s − 4·69-s + 12·71-s + 2·73-s − 10·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s + 1/3·9-s + 0.277·13-s + 1.21·17-s − 1.37·19-s + 0.872·21-s − 0.417·23-s − 0.769·27-s − 1.67·29-s − 0.359·31-s − 0.493·37-s + 0.320·39-s − 0.780·41-s − 0.291·47-s − 3/7·49-s + 1.40·51-s + 1.23·53-s − 1.58·57-s − 1.04·59-s − 0.768·61-s + 0.251·63-s + 0.244·67-s − 0.481·69-s + 1.42·71-s + 0.234·73-s − 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.161310484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.161310484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.21898013419236, −12.62297180505495, −12.31212206259445, −11.61825714402629, −11.19120724576344, −10.80497967275687, −10.13210551048971, −9.786958341985207, −9.194680924077595, −8.679815703042301, −8.410133236260648, −7.922853560292774, −7.464948022724833, −7.088525028718300, −6.182272993429191, −5.885929869848871, −5.200113469712902, −4.729471606101651, −3.946814793874736, −3.639391607055956, −3.140748904891058, −2.369848535041106, −1.857637468639349, −1.516369418391084, −0.4334031027792898,
0.4334031027792898, 1.516369418391084, 1.857637468639349, 2.369848535041106, 3.140748904891058, 3.639391607055956, 3.946814793874736, 4.729471606101651, 5.200113469712902, 5.885929869848871, 6.182272993429191, 7.088525028718300, 7.464948022724833, 7.922853560292774, 8.410133236260648, 8.679815703042301, 9.194680924077595, 9.786958341985207, 10.13210551048971, 10.80497967275687, 11.19120724576344, 11.61825714402629, 12.31212206259445, 12.62297180505495, 13.21898013419236
