| L(s) = 1 | + 2.23·2-s + 2.98·4-s − 2.48·5-s − 7-s + 2.19·8-s − 5.53·10-s + 4.84·11-s − 0.271·13-s − 2.23·14-s − 1.06·16-s − 0.603·17-s + 2.73·19-s − 7.40·20-s + 10.8·22-s − 8.55·23-s + 1.15·25-s − 0.606·26-s − 2.98·28-s + 7.31·29-s + 6.33·31-s − 6.77·32-s − 1.34·34-s + 2.48·35-s − 8.93·37-s + 6.09·38-s − 5.44·40-s + 6.46·41-s + ⋯ |
| L(s) = 1 | + 1.57·2-s + 1.49·4-s − 1.10·5-s − 0.377·7-s + 0.776·8-s − 1.75·10-s + 1.46·11-s − 0.0753·13-s − 0.596·14-s − 0.266·16-s − 0.146·17-s + 0.626·19-s − 1.65·20-s + 2.30·22-s − 1.78·23-s + 0.230·25-s − 0.118·26-s − 0.563·28-s + 1.35·29-s + 1.13·31-s − 1.19·32-s − 0.231·34-s + 0.419·35-s − 1.46·37-s + 0.989·38-s − 0.860·40-s + 1.00·41-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
| \(L(1)\) |
\(\approx\) |
\(4.065512865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.065512865\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.23T + 2T^{2} \) |
| 5 | \( 1 + 2.48T + 5T^{2} \) |
| 11 | \( 1 - 4.84T + 11T^{2} \) |
| 13 | \( 1 + 0.271T + 13T^{2} \) |
| 17 | \( 1 + 0.603T + 17T^{2} \) |
| 19 | \( 1 - 2.73T + 19T^{2} \) |
| 23 | \( 1 + 8.55T + 23T^{2} \) |
| 29 | \( 1 - 7.31T + 29T^{2} \) |
| 31 | \( 1 - 6.33T + 31T^{2} \) |
| 37 | \( 1 + 8.93T + 37T^{2} \) |
| 41 | \( 1 - 6.46T + 41T^{2} \) |
| 43 | \( 1 - 6.83T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 3.31T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 9.40T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 8.79T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 0.343T + 83T^{2} \) |
| 89 | \( 1 - 1.84T + 89T^{2} \) |
| 97 | \( 1 + 4.97T + 97T^{2} \) |
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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
−7.49232369971624008981915582364, −7.00742326562932399126656231247, −6.22596496665798250241296774462, −5.82185812360797018793736595517, −4.78129599306085851496719535505, −4.06706721815404603140136911560, −3.88398618722764632058367151289, −3.04235406010426399230230304379, −2.15136016648177539858647178765, −0.77807337219912186035654991719,
0.77807337219912186035654991719, 2.15136016648177539858647178765, 3.04235406010426399230230304379, 3.88398618722764632058367151289, 4.06706721815404603140136911560, 4.78129599306085851496719535505, 5.82185812360797018793736595517, 6.22596496665798250241296774462, 7.00742326562932399126656231247, 7.49232369971624008981915582364
