| L(s) = 1 | + 1.96·2-s + 1.87·4-s − 2.35·5-s − 7-s − 0.254·8-s − 4.62·10-s + 2.89·11-s + 6.85·13-s − 1.96·14-s − 4.24·16-s − 2.39·17-s − 4.06·19-s − 4.39·20-s + 5.68·22-s + 2.32·23-s + 0.527·25-s + 13.4·26-s − 1.87·28-s − 3.14·29-s − 6.35·31-s − 7.83·32-s − 4.72·34-s + 2.35·35-s + 5.72·37-s − 7.99·38-s + 0.597·40-s − 5.44·41-s + ⋯ |
| L(s) = 1 | + 1.39·2-s + 0.935·4-s − 1.05·5-s − 0.377·7-s − 0.0898·8-s − 1.46·10-s + 0.871·11-s + 1.90·13-s − 0.525·14-s − 1.06·16-s − 0.582·17-s − 0.932·19-s − 0.983·20-s + 1.21·22-s + 0.485·23-s + 0.105·25-s + 2.64·26-s − 0.353·28-s − 0.583·29-s − 1.14·31-s − 1.38·32-s − 0.809·34-s + 0.397·35-s + 0.941·37-s − 1.29·38-s + 0.0945·40-s − 0.850·41-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 | \( 1 - 1.96T + 2T^{2} \) |
| 5 | \( 1 + 2.35T + 5T^{2} \) |
| 11 | \( 1 - 2.89T + 11T^{2} \) |
| 13 | \( 1 - 6.85T + 13T^{2} \) |
| 17 | \( 1 + 2.39T + 17T^{2} \) |
| 19 | \( 1 + 4.06T + 19T^{2} \) |
| 23 | \( 1 - 2.32T + 23T^{2} \) |
| 29 | \( 1 + 3.14T + 29T^{2} \) |
| 31 | \( 1 + 6.35T + 31T^{2} \) |
| 37 | \( 1 - 5.72T + 37T^{2} \) |
| 41 | \( 1 + 5.44T + 41T^{2} \) |
| 43 | \( 1 + 4.92T + 43T^{2} \) |
| 47 | \( 1 - 6.42T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 - 4.10T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 9.78T + 67T^{2} \) |
| 71 | \( 1 - 0.218T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 + 2.97T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 7.20T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−7.07434587648783584617363843935, −6.74082922830440185667898273111, −5.94703930184384305160018229317, −5.45832853219841290884146030208, −4.22276918495888612872326753132, −4.03300686725763147236240817836, −3.54362047505118342680729175169, −2.61173860363575556394213836867, −1.41046763309321373277220694462, 0,
1.41046763309321373277220694462, 2.61173860363575556394213836867, 3.54362047505118342680729175169, 4.03300686725763147236240817836, 4.22276918495888612872326753132, 5.45832853219841290884146030208, 5.94703930184384305160018229317, 6.74082922830440185667898273111, 7.07434587648783584617363843935
