| L(s) = 1 | − 1.72·2-s + 0.961·4-s + 4.11·5-s − 7-s + 1.78·8-s − 7.08·10-s − 1.90·11-s − 0.956·13-s + 1.72·14-s − 4.99·16-s − 0.151·17-s − 2.06·19-s + 3.95·20-s + 3.27·22-s + 6.66·23-s + 11.9·25-s + 1.64·26-s − 0.961·28-s − 4.22·29-s − 2.46·31-s + 5.02·32-s + 0.261·34-s − 4.11·35-s + 0.302·37-s + 3.55·38-s + 7.35·40-s + 6.55·41-s + ⋯ |
| L(s) = 1 | − 1.21·2-s + 0.480·4-s + 1.84·5-s − 0.377·7-s + 0.631·8-s − 2.23·10-s − 0.573·11-s − 0.265·13-s + 0.459·14-s − 1.24·16-s − 0.0367·17-s − 0.473·19-s + 0.884·20-s + 0.698·22-s + 1.38·23-s + 2.38·25-s + 0.322·26-s − 0.181·28-s − 0.784·29-s − 0.442·31-s + 0.888·32-s + 0.0447·34-s − 0.695·35-s + 0.0497·37-s + 0.576·38-s + 1.16·40-s + 1.02·41-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
| \(L(1)\) |
\(\approx\) |
\(1.287596747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.287596747\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.72T + 2T^{2} \) |
| 5 | \( 1 - 4.11T + 5T^{2} \) |
| 11 | \( 1 + 1.90T + 11T^{2} \) |
| 13 | \( 1 + 0.956T + 13T^{2} \) |
| 17 | \( 1 + 0.151T + 17T^{2} \) |
| 19 | \( 1 + 2.06T + 19T^{2} \) |
| 23 | \( 1 - 6.66T + 23T^{2} \) |
| 29 | \( 1 + 4.22T + 29T^{2} \) |
| 31 | \( 1 + 2.46T + 31T^{2} \) |
| 37 | \( 1 - 0.302T + 37T^{2} \) |
| 41 | \( 1 - 6.55T + 41T^{2} \) |
| 43 | \( 1 - 1.79T + 43T^{2} \) |
| 47 | \( 1 - 0.645T + 47T^{2} \) |
| 53 | \( 1 + 0.596T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 6.20T + 71T^{2} \) |
| 73 | \( 1 + 2.03T + 73T^{2} \) |
| 79 | \( 1 - 8.04T + 79T^{2} \) |
| 83 | \( 1 - 9.02T + 83T^{2} \) |
| 89 | \( 1 + 2.23T + 89T^{2} \) |
| 97 | \( 1 - 2.48T + 97T^{2} \) |
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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
−7.88770329358965087750605489790, −7.24133449905183883188785586430, −6.55954548469341110078900492890, −5.87569256319206561090263749009, −5.17945477318495789305052496300, −4.50234642024192903999840974301, −3.14003191916148465520654560271, −2.31852136372198667536447528471, −1.69290512249538641957019041604, −0.68783828466927611260288613739,
0.68783828466927611260288613739, 1.69290512249538641957019041604, 2.31852136372198667536447528471, 3.14003191916148465520654560271, 4.50234642024192903999840974301, 5.17945477318495789305052496300, 5.87569256319206561090263749009, 6.55954548469341110078900492890, 7.24133449905183883188785586430, 7.88770329358965087750605489790
