| L(s) = 1 | − 2.39·3-s − 0.391·5-s + 1.71·7-s + 2.71·9-s + 11-s + 3.71·13-s + 0.935·15-s + 5.43·17-s + 19-s − 4.11·21-s + 3.43·23-s − 4.84·25-s + 0.672·27-s − 3.82·29-s + 3.06·31-s − 2.39·33-s − 0.672·35-s − 4.65·37-s − 8.89·39-s + 1.06·41-s + 3.73·43-s − 1.06·45-s + 8.22·47-s − 4.04·49-s − 13.0·51-s − 1.21·53-s − 0.391·55-s + ⋯ |
| L(s) = 1 | − 1.38·3-s − 0.175·5-s + 0.649·7-s + 0.906·9-s + 0.301·11-s + 1.03·13-s + 0.241·15-s + 1.31·17-s + 0.229·19-s − 0.896·21-s + 0.716·23-s − 0.969·25-s + 0.129·27-s − 0.710·29-s + 0.550·31-s − 0.416·33-s − 0.113·35-s − 0.765·37-s − 1.42·39-s + 0.166·41-s + 0.569·43-s − 0.158·45-s + 1.19·47-s − 0.578·49-s − 1.82·51-s − 0.167·53-s − 0.0527·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.283668615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.283668615\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.39T + 3T^{2} \) |
| 5 | \( 1 + 0.391T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 13 | \( 1 - 3.71T + 13T^{2} \) |
| 17 | \( 1 - 5.43T + 17T^{2} \) |
| 23 | \( 1 - 3.43T + 23T^{2} \) |
| 29 | \( 1 + 3.82T + 29T^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 + 4.65T + 37T^{2} \) |
| 41 | \( 1 - 1.06T + 41T^{2} \) |
| 43 | \( 1 - 3.73T + 43T^{2} \) |
| 47 | \( 1 - 8.22T + 47T^{2} \) |
| 53 | \( 1 + 1.21T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 1.71T + 67T^{2} \) |
| 71 | \( 1 + 1.04T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 8.51T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 4.65T + 97T^{2} \) |
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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
−8.547170052827775962307422240449, −7.78131104017795583305714584512, −7.08193322355096224203985932318, −6.12991540436484065056484559665, −5.68129745644486055585598357586, −4.97003544529368039161272103575, −4.10345479062197999712543946284, −3.20931411403683305008746479280, −1.64189500443334797623731937285, −0.77933037032939345648422726885,
0.77933037032939345648422726885, 1.64189500443334797623731937285, 3.20931411403683305008746479280, 4.10345479062197999712543946284, 4.97003544529368039161272103575, 5.68129745644486055585598357586, 6.12991540436484065056484559665, 7.08193322355096224203985932318, 7.78131104017795583305714584512, 8.547170052827775962307422240449
