| L(s) = 1 | − 2.78·2-s + 5.76·4-s + 0.0591·5-s + 0.765·7-s − 10.4·8-s − 0.164·10-s − 11-s − 1.96·13-s − 2.13·14-s + 17.7·16-s + 4.63·17-s + 6.38·19-s + 0.341·20-s + 2.78·22-s + 1.06·23-s − 4.99·25-s + 5.48·26-s + 4.41·28-s − 3.08·29-s + 4.58·31-s − 28.3·32-s − 12.9·34-s + 0.0452·35-s + 7.04·37-s − 17.7·38-s − 0.621·40-s − 9.41·41-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 2.88·4-s + 0.0264·5-s + 0.289·7-s − 3.71·8-s − 0.0521·10-s − 0.301·11-s − 0.545·13-s − 0.569·14-s + 4.42·16-s + 1.12·17-s + 1.46·19-s + 0.0763·20-s + 0.594·22-s + 0.221·23-s − 0.999·25-s + 1.07·26-s + 0.833·28-s − 0.572·29-s + 0.824·31-s − 5.01·32-s − 2.21·34-s + 0.00765·35-s + 1.15·37-s − 2.88·38-s − 0.0982·40-s − 1.46·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.78T + 2T^{2} \) |
| 5 | \( 1 - 0.0591T + 5T^{2} \) |
| 7 | \( 1 - 0.765T + 7T^{2} \) |
| 13 | \( 1 + 1.96T + 13T^{2} \) |
| 17 | \( 1 - 4.63T + 17T^{2} \) |
| 19 | \( 1 - 6.38T + 19T^{2} \) |
| 23 | \( 1 - 1.06T + 23T^{2} \) |
| 29 | \( 1 + 3.08T + 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 - 7.04T + 37T^{2} \) |
| 41 | \( 1 + 9.41T + 41T^{2} \) |
| 43 | \( 1 + 4.48T + 43T^{2} \) |
| 47 | \( 1 + 0.877T + 47T^{2} \) |
| 53 | \( 1 - 3.43T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 8.42T + 61T^{2} \) |
| 67 | \( 1 + 1.33T + 67T^{2} \) |
| 71 | \( 1 + 8.72T + 71T^{2} \) |
| 73 | \( 1 + 4.74T + 73T^{2} \) |
| 79 | \( 1 - 9.49T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 2.33T + 89T^{2} \) |
| 97 | \( 1 - 3.06T + 97T^{2} \) |
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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
−7.68606689338809028523722703305, −7.26098908118206061950874235817, −6.31612454814770353634696336263, −5.71902038341945049420463946782, −4.89030278920885582980323450799, −3.38275451290132422555471220115, −2.86849271709896302450196153467, −1.82266586954720733712801561102, −1.14024111770002992455196180954, 0,
1.14024111770002992455196180954, 1.82266586954720733712801561102, 2.86849271709896302450196153467, 3.38275451290132422555471220115, 4.89030278920885582980323450799, 5.71902038341945049420463946782, 6.31612454814770353634696336263, 7.26098908118206061950874235817, 7.68606689338809028523722703305
