| L(s) = 1 | − 2-s + 4-s − 0.502·5-s + 3.57·7-s − 8-s + 0.502·10-s + 1.60·11-s + 3.22·13-s − 3.57·14-s + 16-s + 3.44·17-s + 6.49·19-s − 0.502·20-s − 1.60·22-s + 8.18·23-s − 4.74·25-s − 3.22·26-s + 3.57·28-s − 9.85·29-s + 7.98·31-s − 32-s − 3.44·34-s − 1.79·35-s + 10.5·37-s − 6.49·38-s + 0.502·40-s + 7.66·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.224·5-s + 1.35·7-s − 0.353·8-s + 0.158·10-s + 0.484·11-s + 0.894·13-s − 0.956·14-s + 0.250·16-s + 0.834·17-s + 1.48·19-s − 0.112·20-s − 0.342·22-s + 1.70·23-s − 0.949·25-s − 0.632·26-s + 0.676·28-s − 1.82·29-s + 1.43·31-s − 0.176·32-s − 0.590·34-s − 0.303·35-s + 1.72·37-s − 1.05·38-s + 0.0794·40-s + 1.19·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.026282812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.026282812\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 269 | \( 1 - T \) |
| good | 5 | \( 1 + 0.502T + 5T^{2} \) |
| 7 | \( 1 - 3.57T + 7T^{2} \) |
| 11 | \( 1 - 1.60T + 11T^{2} \) |
| 13 | \( 1 - 3.22T + 13T^{2} \) |
| 17 | \( 1 - 3.44T + 17T^{2} \) |
| 19 | \( 1 - 6.49T + 19T^{2} \) |
| 23 | \( 1 - 8.18T + 23T^{2} \) |
| 29 | \( 1 + 9.85T + 29T^{2} \) |
| 31 | \( 1 - 7.98T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 7.66T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 + 4.01T + 47T^{2} \) |
| 53 | \( 1 - 4.13T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 4.52T + 67T^{2} \) |
| 71 | \( 1 - 5.02T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 9.91T + 83T^{2} \) |
| 89 | \( 1 + 5.92T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.158496223403852067142731354908, −7.69814678911809211939019378485, −7.15458890303952021478323026296, −6.07311736299047427569793422569, −5.45640715460226308584245338683, −4.59304371418740254730880758612, −3.66935466367683659035614718553, −2.79897082541813186773361972724, −1.50563926244994584265841602692, −1.01741431158473015567995876812,
1.01741431158473015567995876812, 1.50563926244994584265841602692, 2.79897082541813186773361972724, 3.66935466367683659035614718553, 4.59304371418740254730880758612, 5.45640715460226308584245338683, 6.07311736299047427569793422569, 7.15458890303952021478323026296, 7.69814678911809211939019378485, 8.158496223403852067142731354908
