| L(s) = 1 | + 0.336·3-s + 3.95·5-s + 7-s − 2.88·9-s + 11-s + 13-s + 1.33·15-s − 0.794·17-s + 4.35·19-s + 0.336·21-s − 1.29·23-s + 10.6·25-s − 1.98·27-s + 3.76·29-s + 6.69·31-s + 0.336·33-s + 3.95·35-s − 2.72·37-s + 0.336·39-s − 3.50·41-s − 1.63·43-s − 11.4·45-s − 0.830·47-s + 49-s − 0.267·51-s + 10.8·53-s + 3.95·55-s + ⋯ |
| L(s) = 1 | + 0.194·3-s + 1.76·5-s + 0.377·7-s − 0.962·9-s + 0.301·11-s + 0.277·13-s + 0.344·15-s − 0.192·17-s + 1.00·19-s + 0.0735·21-s − 0.270·23-s + 2.13·25-s − 0.381·27-s + 0.699·29-s + 1.20·31-s + 0.0586·33-s + 0.668·35-s − 0.447·37-s + 0.0539·39-s − 0.547·41-s − 0.249·43-s − 1.70·45-s − 0.121·47-s + 0.142·49-s − 0.0374·51-s + 1.49·53-s + 0.533·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.480946762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.480946762\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.336T + 3T^{2} \) |
| 5 | \( 1 - 3.95T + 5T^{2} \) |
| 17 | \( 1 + 0.794T + 17T^{2} \) |
| 19 | \( 1 - 4.35T + 19T^{2} \) |
| 23 | \( 1 + 1.29T + 23T^{2} \) |
| 29 | \( 1 - 3.76T + 29T^{2} \) |
| 31 | \( 1 - 6.69T + 31T^{2} \) |
| 37 | \( 1 + 2.72T + 37T^{2} \) |
| 41 | \( 1 + 3.50T + 41T^{2} \) |
| 43 | \( 1 + 1.63T + 43T^{2} \) |
| 47 | \( 1 + 0.830T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 5.39T + 59T^{2} \) |
| 61 | \( 1 - 3.96T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 4.19T + 71T^{2} \) |
| 73 | \( 1 - 5.43T + 73T^{2} \) |
| 79 | \( 1 + 9.39T + 79T^{2} \) |
| 83 | \( 1 - 5.82T + 83T^{2} \) |
| 89 | \( 1 + 5.81T + 89T^{2} \) |
| 97 | \( 1 + 3.83T + 97T^{2} \) |
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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
−7.973251853102695885287917750842, −6.95592666155921740256681850804, −6.38419223354668293084997693753, −5.66996451824406754389123888459, −5.29791565412362581604840285586, −4.41981019016074931964654057707, −3.25505280364672381835019629902, −2.60673849621687912385760456508, −1.83892675526679344647513889590, −0.957493290952404229500295800345,
0.957493290952404229500295800345, 1.83892675526679344647513889590, 2.60673849621687912385760456508, 3.25505280364672381835019629902, 4.41981019016074931964654057707, 5.29791565412362581604840285586, 5.66996451824406754389123888459, 6.38419223354668293084997693753, 6.95592666155921740256681850804, 7.973251853102695885287917750842
