| L(s) = 1 | + 2.77·3-s + 2.31·7-s + 4.67·9-s + 2.16·11-s + 6.25·13-s + 7.10·17-s − 19-s + 6.40·21-s − 8.99·23-s + 4.63·27-s + 3.16·29-s − 9.95·31-s + 5.98·33-s + 9.43·37-s + 17.3·39-s − 10.8·41-s + 1.05·43-s − 6.98·47-s − 1.66·49-s + 19.6·51-s − 2.69·53-s − 2.77·57-s + 11.2·59-s − 4.36·61-s + 10.7·63-s + 6.25·67-s − 24.9·69-s + ⋯ |
| L(s) = 1 | + 1.59·3-s + 0.873·7-s + 1.55·9-s + 0.651·11-s + 1.73·13-s + 1.72·17-s − 0.229·19-s + 1.39·21-s − 1.87·23-s + 0.892·27-s + 0.587·29-s − 1.78·31-s + 1.04·33-s + 1.55·37-s + 2.77·39-s − 1.68·41-s + 0.160·43-s − 1.01·47-s − 0.237·49-s + 2.75·51-s − 0.370·53-s − 0.366·57-s + 1.45·59-s − 0.558·61-s + 1.36·63-s + 0.764·67-s − 3.00·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.458366608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.458366608\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.77T + 3T^{2} \) |
| 7 | \( 1 - 2.31T + 7T^{2} \) |
| 11 | \( 1 - 2.16T + 11T^{2} \) |
| 13 | \( 1 - 6.25T + 13T^{2} \) |
| 17 | \( 1 - 7.10T + 17T^{2} \) |
| 23 | \( 1 + 8.99T + 23T^{2} \) |
| 29 | \( 1 - 3.16T + 29T^{2} \) |
| 31 | \( 1 + 9.95T + 31T^{2} \) |
| 37 | \( 1 - 9.43T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 1.05T + 43T^{2} \) |
| 47 | \( 1 + 6.98T + 47T^{2} \) |
| 53 | \( 1 + 2.69T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 4.36T + 61T^{2} \) |
| 67 | \( 1 - 6.25T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 8.64T + 73T^{2} \) |
| 79 | \( 1 + 9.93T + 79T^{2} \) |
| 83 | \( 1 + 9.82T + 83T^{2} \) |
| 89 | \( 1 + 1.63T + 89T^{2} \) |
| 97 | \( 1 - 9.27T + 97T^{2} \) |
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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
−8.416844845775407031458014684867, −8.003322234917908542258292407072, −7.37882186968181741335174974399, −6.27974157867221583863041133764, −5.58404774455847619878433390470, −4.34622309274093397128652057411, −3.71298704342400990336202592685, −3.15702560461685351166608566606, −1.86132197103629459089971158830, −1.36995978014762498193570884767,
1.36995978014762498193570884767, 1.86132197103629459089971158830, 3.15702560461685351166608566606, 3.71298704342400990336202592685, 4.34622309274093397128652057411, 5.58404774455847619878433390470, 6.27974157867221583863041133764, 7.37882186968181741335174974399, 8.003322234917908542258292407072, 8.416844845775407031458014684867
