| L(s) = 1 | − 2.30·3-s + 1.38·5-s + 2.51·7-s + 2.29·9-s − 1.87·11-s + 0.839·13-s − 3.18·15-s + 5.56·17-s − 2.82·19-s − 5.79·21-s − 6.93·23-s − 3.08·25-s + 1.62·27-s + 3.43·29-s + 5.09·31-s + 4.31·33-s + 3.48·35-s − 0.844·37-s − 1.93·39-s − 5.00·41-s + 3.73·43-s + 3.17·45-s − 12.2·47-s − 0.649·49-s − 12.7·51-s + 6.38·53-s − 2.59·55-s + ⋯ |
| L(s) = 1 | − 1.32·3-s + 0.619·5-s + 0.952·7-s + 0.765·9-s − 0.564·11-s + 0.232·13-s − 0.822·15-s + 1.34·17-s − 0.648·19-s − 1.26·21-s − 1.44·23-s − 0.616·25-s + 0.311·27-s + 0.637·29-s + 0.914·31-s + 0.750·33-s + 0.589·35-s − 0.138·37-s − 0.309·39-s − 0.780·41-s + 0.569·43-s + 0.473·45-s − 1.78·47-s − 0.0928·49-s − 1.79·51-s + 0.877·53-s − 0.349·55-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 | \( 1 + 2.30T + 3T^{2} \) |
| 5 | \( 1 - 1.38T + 5T^{2} \) |
| 7 | \( 1 - 2.51T + 7T^{2} \) |
| 11 | \( 1 + 1.87T + 11T^{2} \) |
| 13 | \( 1 - 0.839T + 13T^{2} \) |
| 17 | \( 1 - 5.56T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 6.93T + 23T^{2} \) |
| 29 | \( 1 - 3.43T + 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 + 0.844T + 37T^{2} \) |
| 41 | \( 1 + 5.00T + 41T^{2} \) |
| 43 | \( 1 - 3.73T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 6.38T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 2.24T + 61T^{2} \) |
| 67 | \( 1 + 6.07T + 67T^{2} \) |
| 71 | \( 1 - 1.57T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 0.530T + 89T^{2} \) |
| 97 | \( 1 - 0.656T + 97T^{2} \) |
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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
−7.54256472611237722410850065952, −6.46001403558402487735492896386, −6.05420875761629983300477375941, −5.44887610133067521952953192802, −4.87896261258901019948507849444, −4.19822276098535773877969842850, −3.05769442121662340156131066374, −1.96735947934547499483625036088, −1.22486588550571392112041487112, 0,
1.22486588550571392112041487112, 1.96735947934547499483625036088, 3.05769442121662340156131066374, 4.19822276098535773877969842850, 4.87896261258901019948507849444, 5.44887610133067521952953192802, 6.05420875761629983300477375941, 6.46001403558402487735492896386, 7.54256472611237722410850065952
