| L(s) = 1 | − 3.60·5-s + 3.60·7-s − 11-s + 4·13-s − 7.21·19-s + 6·23-s + 7.99·25-s + 7.21·29-s − 3.60·31-s − 12.9·35-s + 10·37-s + 7.21·41-s + 7.21·43-s + 10·47-s + 5.99·49-s + 3.60·53-s + 3.60·55-s + 4·59-s − 14.4·65-s + 7.21·67-s − 8·71-s − 3·73-s − 3.60·77-s − 14.4·79-s − 9·83-s − 7.21·89-s + 14.4·91-s + ⋯ |
| L(s) = 1 | − 1.61·5-s + 1.36·7-s − 0.301·11-s + 1.10·13-s − 1.65·19-s + 1.25·23-s + 1.59·25-s + 1.33·29-s − 0.647·31-s − 2.19·35-s + 1.64·37-s + 1.12·41-s + 1.09·43-s + 1.45·47-s + 0.857·49-s + 0.495·53-s + 0.486·55-s + 0.520·59-s − 1.78·65-s + 0.880·67-s − 0.949·71-s − 0.351·73-s − 0.410·77-s − 1.62·79-s − 0.987·83-s − 0.764·89-s + 1.51·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.365812731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.365812731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.60T + 5T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 7.21T + 29T^{2} \) |
| 31 | \( 1 + 3.60T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 7.21T + 41T^{2} \) |
| 43 | \( 1 - 7.21T + 43T^{2} \) |
| 47 | \( 1 - 10T + 47T^{2} \) |
| 53 | \( 1 - 3.60T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 7.21T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 3T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 9T + 83T^{2} \) |
| 89 | \( 1 + 7.21T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−10.61329130962769716821788550023, −8.928035848704536448691635188715, −8.393739727683504466634359674695, −7.78611273786181910275629725402, −6.95068919942131121681486334724, −5.72670324801817356906859602297, −4.44082148729733690222942083924, −4.13218273163520837512685375835, −2.67852308830331863665695825763, −0.989698206489348970590235414714,
0.989698206489348970590235414714, 2.67852308830331863665695825763, 4.13218273163520837512685375835, 4.44082148729733690222942083924, 5.72670324801817356906859602297, 6.95068919942131121681486334724, 7.78611273786181910275629725402, 8.393739727683504466634359674695, 8.928035848704536448691635188715, 10.61329130962769716821788550023
