| L(s) = 1 | − 1.84·2-s + 1.41·4-s + 3.49·5-s − 7-s + 1.08·8-s − 6.45·10-s − 1.44·11-s + 0.419·13-s + 1.84·14-s − 4.82·16-s + 3.49·17-s − 3.81·19-s + 4.94·20-s + 2.67·22-s − 7.19·23-s + 7.19·25-s − 0.774·26-s − 1.41·28-s − 1.43·29-s + 4.42·31-s + 6.75·32-s − 6.46·34-s − 3.49·35-s + 5.04·37-s + 7.04·38-s + 3.77·40-s − 9.79·41-s + ⋯ |
| L(s) = 1 | − 1.30·2-s + 0.707·4-s + 1.56·5-s − 0.377·7-s + 0.382·8-s − 2.04·10-s − 0.436·11-s + 0.116·13-s + 0.493·14-s − 1.20·16-s + 0.847·17-s − 0.874·19-s + 1.10·20-s + 0.570·22-s − 1.50·23-s + 1.43·25-s − 0.151·26-s − 0.267·28-s − 0.265·29-s + 0.794·31-s + 1.19·32-s − 1.10·34-s − 0.590·35-s + 0.829·37-s + 1.14·38-s + 0.596·40-s − 1.52·41-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 | \( 1 + 1.84T + 2T^{2} \) |
| 5 | \( 1 - 3.49T + 5T^{2} \) |
| 11 | \( 1 + 1.44T + 11T^{2} \) |
| 13 | \( 1 - 0.419T + 13T^{2} \) |
| 17 | \( 1 - 3.49T + 17T^{2} \) |
| 19 | \( 1 + 3.81T + 19T^{2} \) |
| 23 | \( 1 + 7.19T + 23T^{2} \) |
| 29 | \( 1 + 1.43T + 29T^{2} \) |
| 31 | \( 1 - 4.42T + 31T^{2} \) |
| 37 | \( 1 - 5.04T + 37T^{2} \) |
| 41 | \( 1 + 9.79T + 41T^{2} \) |
| 43 | \( 1 + 4.63T + 43T^{2} \) |
| 47 | \( 1 - 6.75T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 7.12T + 59T^{2} \) |
| 61 | \( 1 + 0.158T + 61T^{2} \) |
| 67 | \( 1 - 0.583T + 67T^{2} \) |
| 71 | \( 1 - 9.97T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−7.899672700138229010327713636250, −6.67714595732105667012680909176, −6.37633982245748926527442844264, −5.56267532560248079876148834854, −4.85551564295858509489413974371, −3.81802435264555733400081043330, −2.65450289008237725206806261563, −1.99535073112783139247900384955, −1.24917257554006876973528623561, 0,
1.24917257554006876973528623561, 1.99535073112783139247900384955, 2.65450289008237725206806261563, 3.81802435264555733400081043330, 4.85551564295858509489413974371, 5.56267532560248079876148834854, 6.37633982245748926527442844264, 6.67714595732105667012680909176, 7.899672700138229010327713636250
