| L(s) = 1 | + 2.30·2-s + 3.30·4-s + 1.35·5-s − 7-s + 2.99·8-s + 3.12·10-s + 1.22·11-s − 5.64·13-s − 2.30·14-s + 0.300·16-s − 6.78·17-s + 2.18·19-s + 4.47·20-s + 2.80·22-s − 5.37·23-s − 3.15·25-s − 13.0·26-s − 3.30·28-s − 0.967·29-s + 3.27·31-s − 5.30·32-s − 15.6·34-s − 1.35·35-s + 10.3·37-s + 5.03·38-s + 4.06·40-s − 8.37·41-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.65·4-s + 0.606·5-s − 0.377·7-s + 1.06·8-s + 0.987·10-s + 0.367·11-s − 1.56·13-s − 0.615·14-s + 0.0750·16-s − 1.64·17-s + 0.501·19-s + 1.00·20-s + 0.599·22-s − 1.12·23-s − 0.631·25-s − 2.55·26-s − 0.624·28-s − 0.179·29-s + 0.588·31-s − 0.937·32-s − 2.67·34-s − 0.229·35-s + 1.69·37-s + 0.816·38-s + 0.643·40-s − 1.30·41-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 | \( 1 - 2.30T + 2T^{2} \) |
| 5 | \( 1 - 1.35T + 5T^{2} \) |
| 11 | \( 1 - 1.22T + 11T^{2} \) |
| 13 | \( 1 + 5.64T + 13T^{2} \) |
| 17 | \( 1 + 6.78T + 17T^{2} \) |
| 19 | \( 1 - 2.18T + 19T^{2} \) |
| 23 | \( 1 + 5.37T + 23T^{2} \) |
| 29 | \( 1 + 0.967T + 29T^{2} \) |
| 31 | \( 1 - 3.27T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 8.37T + 41T^{2} \) |
| 43 | \( 1 + 3.70T + 43T^{2} \) |
| 47 | \( 1 - 6.13T + 47T^{2} \) |
| 53 | \( 1 + 3.82T + 53T^{2} \) |
| 59 | \( 1 + 5.08T + 59T^{2} \) |
| 61 | \( 1 + 9.24T + 61T^{2} \) |
| 67 | \( 1 + 5.49T + 67T^{2} \) |
| 71 | \( 1 + 7.44T + 71T^{2} \) |
| 73 | \( 1 - 1.22T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 9.75T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 + 5.03T + 97T^{2} \) |
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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
−7.19808660895016089114121089734, −6.43201320461150198193780205986, −6.11223327294041288622156121735, −5.29403410985353034887445359710, −4.58859307450374962752348604001, −4.15710383303215360168409373619, −3.15103847565396668956517460993, −2.43609409006422828614666826536, −1.85712466030492440425469769987, 0,
1.85712466030492440425469769987, 2.43609409006422828614666826536, 3.15103847565396668956517460993, 4.15710383303215360168409373619, 4.58859307450374962752348604001, 5.29403410985353034887445359710, 6.11223327294041288622156121735, 6.43201320461150198193780205986, 7.19808660895016089114121089734
