| L(s) = 1 | + 2.42·2-s + 3.88·4-s − 2.61·5-s + 7-s + 4.56·8-s − 6.34·10-s + 4.90·11-s + 2.57·13-s + 2.42·14-s + 3.30·16-s + 1.92·17-s − 3.81·19-s − 10.1·20-s + 11.8·22-s − 3.20·23-s + 1.84·25-s + 6.25·26-s + 3.88·28-s + 5.89·29-s − 3.10·31-s − 1.10·32-s + 4.66·34-s − 2.61·35-s + 11.9·37-s − 9.24·38-s − 11.9·40-s − 0.926·41-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 1.94·4-s − 1.16·5-s + 0.377·7-s + 1.61·8-s − 2.00·10-s + 1.47·11-s + 0.714·13-s + 0.648·14-s + 0.826·16-s + 0.466·17-s − 0.874·19-s − 2.27·20-s + 2.53·22-s − 0.669·23-s + 0.368·25-s + 1.22·26-s + 0.733·28-s + 1.09·29-s − 0.557·31-s − 0.195·32-s + 0.800·34-s − 0.442·35-s + 1.96·37-s − 1.49·38-s − 1.88·40-s − 0.144·41-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
| \(L(1)\) |
\(\approx\) |
\(5.840752044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.840752044\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.42T + 2T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 11 | \( 1 - 4.90T + 11T^{2} \) |
| 13 | \( 1 - 2.57T + 13T^{2} \) |
| 17 | \( 1 - 1.92T + 17T^{2} \) |
| 19 | \( 1 + 3.81T + 19T^{2} \) |
| 23 | \( 1 + 3.20T + 23T^{2} \) |
| 29 | \( 1 - 5.89T + 29T^{2} \) |
| 31 | \( 1 + 3.10T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 + 0.926T + 41T^{2} \) |
| 43 | \( 1 - 2.93T + 43T^{2} \) |
| 47 | \( 1 - 9.46T + 47T^{2} \) |
| 53 | \( 1 - 6.84T + 53T^{2} \) |
| 59 | \( 1 + 3.42T + 59T^{2} \) |
| 61 | \( 1 + 3.67T + 61T^{2} \) |
| 67 | \( 1 - 8.49T + 67T^{2} \) |
| 71 | \( 1 - 6.86T + 71T^{2} \) |
| 73 | \( 1 + 7.98T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 2.16T + 97T^{2} \) |
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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
−7.57295200035100157681107453282, −6.98117368278525325268576716260, −6.16182133922176427330157892903, −5.85542599140732163129825680013, −4.72056030889088987845877321475, −4.09142070103476813874668931272, −3.93926041516851390026369339039, −3.06055218756960228519104561346, −2.07366394137467016602204149849, −0.958625929251646544754007120257,
0.958625929251646544754007120257, 2.07366394137467016602204149849, 3.06055218756960228519104561346, 3.93926041516851390026369339039, 4.09142070103476813874668931272, 4.72056030889088987845877321475, 5.85542599140732163129825680013, 6.16182133922176427330157892903, 6.98117368278525325268576716260, 7.57295200035100157681107453282
