| L(s) = 1 | + 2.16·2-s + 2.70·4-s + 2.09·5-s − 7-s + 1.53·8-s + 4.54·10-s − 6.57·11-s + 2.39·13-s − 2.16·14-s − 2.08·16-s + 1.17·17-s − 3.96·19-s + 5.66·20-s − 14.2·22-s − 0.848·23-s − 0.616·25-s + 5.19·26-s − 2.70·28-s − 8.50·29-s + 8.20·31-s − 7.59·32-s + 2.55·34-s − 2.09·35-s − 4.04·37-s − 8.59·38-s + 3.21·40-s + 6.47·41-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 1.35·4-s + 0.936·5-s − 0.377·7-s + 0.542·8-s + 1.43·10-s − 1.98·11-s + 0.664·13-s − 0.579·14-s − 0.521·16-s + 0.285·17-s − 0.908·19-s + 1.26·20-s − 3.04·22-s − 0.176·23-s − 0.123·25-s + 1.01·26-s − 0.511·28-s − 1.57·29-s + 1.47·31-s − 1.34·32-s + 0.438·34-s − 0.353·35-s − 0.665·37-s − 1.39·38-s + 0.508·40-s + 1.01·41-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.16T + 2T^{2} \) |
| 5 | \( 1 - 2.09T + 5T^{2} \) |
| 11 | \( 1 + 6.57T + 11T^{2} \) |
| 13 | \( 1 - 2.39T + 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 19 | \( 1 + 3.96T + 19T^{2} \) |
| 23 | \( 1 + 0.848T + 23T^{2} \) |
| 29 | \( 1 + 8.50T + 29T^{2} \) |
| 31 | \( 1 - 8.20T + 31T^{2} \) |
| 37 | \( 1 + 4.04T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + 0.335T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 3.76T + 53T^{2} \) |
| 59 | \( 1 + 4.27T + 59T^{2} \) |
| 61 | \( 1 - 15.6T + 61T^{2} \) |
| 67 | \( 1 + 9.44T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + 5.20T + 89T^{2} \) |
| 97 | \( 1 + 3.10T + 97T^{2} \) |
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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
−7.25369237293536012463382688573, −6.41506619779811345165095254849, −5.87222977482048135384253567775, −5.46167073007058942681703248241, −4.76812587735997669767507928117, −3.98020563493679663172787454135, −3.10664301880236035457693317165, −2.52741462349553412497641531158, −1.77789982156930920251468064004, 0,
1.77789982156930920251468064004, 2.52741462349553412497641531158, 3.10664301880236035457693317165, 3.98020563493679663172787454135, 4.76812587735997669767507928117, 5.46167073007058942681703248241, 5.87222977482048135384253567775, 6.41506619779811345165095254849, 7.25369237293536012463382688573
