| L(s) = 1 | + 2.23·3-s − 3.23·5-s − 0.236·7-s + 2.00·9-s − 0.236·11-s + 3.47·13-s − 7.23·15-s + 2·17-s − 5.70·19-s − 0.527·21-s + 2.23·23-s + 5.47·25-s − 2.23·27-s − 7.23·29-s + 5.70·31-s − 0.527·33-s + 0.763·35-s + 4.47·37-s + 7.76·39-s − 0.763·41-s − 5.76·43-s − 6.47·45-s + 0.236·47-s − 6.94·49-s + 4.47·51-s + 8.47·53-s + 0.763·55-s + ⋯ |
| L(s) = 1 | + 1.29·3-s − 1.44·5-s − 0.0892·7-s + 0.666·9-s − 0.0711·11-s + 0.962·13-s − 1.86·15-s + 0.485·17-s − 1.30·19-s − 0.115·21-s + 0.466·23-s + 1.09·25-s − 0.430·27-s − 1.34·29-s + 1.02·31-s − 0.0918·33-s + 0.129·35-s + 0.735·37-s + 1.24·39-s − 0.119·41-s − 0.878·43-s − 0.964·45-s + 0.0344·47-s − 0.992·49-s + 0.626·51-s + 1.16·53-s + 0.103·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 + 0.236T + 7T^{2} \) |
| 11 | \( 1 + 0.236T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 23 | \( 1 - 2.23T + 23T^{2} \) |
| 29 | \( 1 + 7.23T + 29T^{2} \) |
| 31 | \( 1 - 5.70T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 0.763T + 41T^{2} \) |
| 43 | \( 1 + 5.76T + 43T^{2} \) |
| 47 | \( 1 - 0.236T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 1.47T + 61T^{2} \) |
| 67 | \( 1 - 8.70T + 67T^{2} \) |
| 71 | \( 1 + 5.23T + 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 - 4.94T + 79T^{2} \) |
| 83 | \( 1 + 3.29T + 83T^{2} \) |
| 89 | \( 1 + 5.52T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−7.70853080026506351883304197473, −7.02969309791998238552444737320, −6.26966062804210803846889956209, −5.29974653453893358643986619060, −4.19715501768570964061798392760, −3.89420992044347418125940503392, −3.19262924857777904944509575836, −2.46492246749014708356224547771, −1.34075904265623587125103303333, 0,
1.34075904265623587125103303333, 2.46492246749014708356224547771, 3.19262924857777904944509575836, 3.89420992044347418125940503392, 4.19715501768570964061798392760, 5.29974653453893358643986619060, 6.26966062804210803846889956209, 7.02969309791998238552444737320, 7.70853080026506351883304197473
