| L(s) = 1 | − 1.84·2-s + 1.39·4-s − 2.07·5-s + 4.15·7-s + 1.11·8-s + 3.82·10-s − 0.145·11-s − 6.13·13-s − 7.64·14-s − 4.84·16-s + 2.97·17-s + 3.57·19-s − 2.89·20-s + 0.268·22-s − 23-s − 0.685·25-s + 11.3·26-s + 5.78·28-s + 29-s + 5.64·31-s + 6.69·32-s − 5.47·34-s − 8.62·35-s − 8.51·37-s − 6.58·38-s − 2.31·40-s + 5.62·41-s + ⋯ |
| L(s) = 1 | − 1.30·2-s + 0.696·4-s − 0.928·5-s + 1.56·7-s + 0.394·8-s + 1.21·10-s − 0.0439·11-s − 1.70·13-s − 2.04·14-s − 1.21·16-s + 0.721·17-s + 0.820·19-s − 0.647·20-s + 0.0572·22-s − 0.208·23-s − 0.137·25-s + 2.21·26-s + 1.09·28-s + 0.185·29-s + 1.01·31-s + 1.18·32-s − 0.939·34-s − 1.45·35-s − 1.39·37-s − 1.06·38-s − 0.366·40-s + 0.878·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.84T + 2T^{2} \) |
| 5 | \( 1 + 2.07T + 5T^{2} \) |
| 7 | \( 1 - 4.15T + 7T^{2} \) |
| 11 | \( 1 + 0.145T + 11T^{2} \) |
| 13 | \( 1 + 6.13T + 13T^{2} \) |
| 17 | \( 1 - 2.97T + 17T^{2} \) |
| 19 | \( 1 - 3.57T + 19T^{2} \) |
| 31 | \( 1 - 5.64T + 31T^{2} \) |
| 37 | \( 1 + 8.51T + 37T^{2} \) |
| 41 | \( 1 - 5.62T + 41T^{2} \) |
| 43 | \( 1 - 7.38T + 43T^{2} \) |
| 47 | \( 1 + 4.24T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 + 8.75T + 59T^{2} \) |
| 61 | \( 1 + 7.03T + 61T^{2} \) |
| 67 | \( 1 + 4.17T + 67T^{2} \) |
| 71 | \( 1 - 4.12T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 5.33T + 89T^{2} \) |
| 97 | \( 1 + 7.69T + 97T^{2} \) |
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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
−7.86721720083893274317198357344, −7.56416061587040677579933778083, −6.73212528675353446660668310820, −5.40829925379117971710991758411, −4.77549760975074375349868078559, −4.25505167790963306804062998754, −3.00192536511528472541680634552, −1.97942388106573099793231832575, −1.12496966894892085821644761742, 0,
1.12496966894892085821644761742, 1.97942388106573099793231832575, 3.00192536511528472541680634552, 4.25505167790963306804062998754, 4.77549760975074375349868078559, 5.40829925379117971710991758411, 6.73212528675353446660668310820, 7.56416061587040677579933778083, 7.86721720083893274317198357344
