| L(s) = 1 | + 0.981·2-s − 1.03·4-s − 2.59·5-s + 7-s − 2.98·8-s − 2.54·10-s + 0.832·11-s − 5.10·13-s + 0.981·14-s − 0.855·16-s − 7.10·17-s − 7.54·19-s + 2.68·20-s + 0.817·22-s − 8.33·23-s + 1.73·25-s − 5.01·26-s − 1.03·28-s + 8.62·29-s − 6.02·31-s + 5.12·32-s − 6.97·34-s − 2.59·35-s − 5.52·37-s − 7.41·38-s + 7.73·40-s + 8.68·41-s + ⋯ |
| L(s) = 1 | + 0.694·2-s − 0.517·4-s − 1.16·5-s + 0.377·7-s − 1.05·8-s − 0.805·10-s + 0.250·11-s − 1.41·13-s + 0.262·14-s − 0.213·16-s − 1.72·17-s − 1.73·19-s + 0.600·20-s + 0.174·22-s − 1.73·23-s + 0.346·25-s − 0.983·26-s − 0.195·28-s + 1.60·29-s − 1.08·31-s + 0.905·32-s − 1.19·34-s − 0.438·35-s − 0.908·37-s − 1.20·38-s + 1.22·40-s + 1.35·41-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
| \(L(1)\) |
\(\approx\) |
\(0.3149272294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3149272294\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.981T + 2T^{2} \) |
| 5 | \( 1 + 2.59T + 5T^{2} \) |
| 11 | \( 1 - 0.832T + 11T^{2} \) |
| 13 | \( 1 + 5.10T + 13T^{2} \) |
| 17 | \( 1 + 7.10T + 17T^{2} \) |
| 19 | \( 1 + 7.54T + 19T^{2} \) |
| 23 | \( 1 + 8.33T + 23T^{2} \) |
| 29 | \( 1 - 8.62T + 29T^{2} \) |
| 31 | \( 1 + 6.02T + 31T^{2} \) |
| 37 | \( 1 + 5.52T + 37T^{2} \) |
| 41 | \( 1 - 8.68T + 41T^{2} \) |
| 43 | \( 1 - 4.88T + 43T^{2} \) |
| 47 | \( 1 + 2.27T + 47T^{2} \) |
| 53 | \( 1 + 3.81T + 53T^{2} \) |
| 59 | \( 1 - 3.88T + 59T^{2} \) |
| 61 | \( 1 + 7.64T + 61T^{2} \) |
| 67 | \( 1 - 9.03T + 67T^{2} \) |
| 71 | \( 1 - 9.47T + 71T^{2} \) |
| 73 | \( 1 + 6.75T + 73T^{2} \) |
| 79 | \( 1 - 5.46T + 79T^{2} \) |
| 83 | \( 1 + 9.63T + 83T^{2} \) |
| 89 | \( 1 + 9.34T + 89T^{2} \) |
| 97 | \( 1 + 0.647T + 97T^{2} \) |
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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
−8.053304288741150938764225188020, −7.00978911904032282718567788140, −6.48913982299483425508282052802, −5.60838738298573625484102933039, −4.72175678099691787544826287497, −4.18446401856904285490757614077, −4.02733776976251794273010600891, −2.74720710187737299827494781178, −2.04703652506591994651758585025, −0.23378839832751369666253754405,
0.23378839832751369666253754405, 2.04703652506591994651758585025, 2.74720710187737299827494781178, 4.02733776976251794273010600891, 4.18446401856904285490757614077, 4.72175678099691787544826287497, 5.60838738298573625484102933039, 6.48913982299483425508282052802, 7.00978911904032282718567788140, 8.053304288741150938764225188020
