| L(s) = 1 | + 1.67·2-s + 0.821·4-s + 1.48·5-s + 7-s − 1.97·8-s + 2.49·10-s + 5.84·11-s + 1.01·13-s + 1.67·14-s − 4.96·16-s + 8.08·17-s + 6.65·19-s + 1.21·20-s + 9.81·22-s + 2.54·23-s − 2.80·25-s + 1.70·26-s + 0.821·28-s − 4.36·29-s − 2.49·31-s − 4.38·32-s + 13.5·34-s + 1.48·35-s − 8.29·37-s + 11.1·38-s − 2.93·40-s + 0.441·41-s + ⋯ |
| L(s) = 1 | + 1.18·2-s + 0.410·4-s + 0.663·5-s + 0.377·7-s − 0.699·8-s + 0.787·10-s + 1.76·11-s + 0.282·13-s + 0.448·14-s − 1.24·16-s + 1.96·17-s + 1.52·19-s + 0.272·20-s + 2.09·22-s + 0.531·23-s − 0.560·25-s + 0.335·26-s + 0.155·28-s − 0.810·29-s − 0.447·31-s − 0.775·32-s + 2.32·34-s + 0.250·35-s − 1.36·37-s + 1.81·38-s − 0.464·40-s + 0.0689·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.379510756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.379510756\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 1.67T + 2T^{2} \) |
| 5 | \( 1 - 1.48T + 5T^{2} \) |
| 11 | \( 1 - 5.84T + 11T^{2} \) |
| 13 | \( 1 - 1.01T + 13T^{2} \) |
| 17 | \( 1 - 8.08T + 17T^{2} \) |
| 19 | \( 1 - 6.65T + 19T^{2} \) |
| 23 | \( 1 - 2.54T + 23T^{2} \) |
| 29 | \( 1 + 4.36T + 29T^{2} \) |
| 31 | \( 1 + 2.49T + 31T^{2} \) |
| 37 | \( 1 + 8.29T + 37T^{2} \) |
| 41 | \( 1 - 0.441T + 41T^{2} \) |
| 43 | \( 1 + 1.58T + 43T^{2} \) |
| 47 | \( 1 - 7.63T + 47T^{2} \) |
| 53 | \( 1 - 8.25T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 4.13T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 4.41T + 83T^{2} \) |
| 89 | \( 1 + 1.22T + 89T^{2} \) |
| 97 | \( 1 - 0.556T + 97T^{2} \) |
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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
−7.51556276865322033536366238725, −7.08159741178201439137482507813, −6.07377315040541843383954352469, −5.65346659015276091141831730304, −5.20165960269051034881893359742, −4.21226170698231280515215847637, −3.57123630418000287552246057146, −3.08368636278544864349909997381, −1.77900129995343560477682696618, −1.06695264240014965020843559061,
1.06695264240014965020843559061, 1.77900129995343560477682696618, 3.08368636278544864349909997381, 3.57123630418000287552246057146, 4.21226170698231280515215847637, 5.20165960269051034881893359742, 5.65346659015276091141831730304, 6.07377315040541843383954352469, 7.08159741178201439137482507813, 7.51556276865322033536366238725
