| L(s) = 1 | − 1.53·2-s + 0.347·4-s + 3.22·5-s − 0.652·7-s + 2.53·8-s − 4.94·10-s + 11-s + 2.29·13-s + 0.999·14-s − 4.57·16-s − 2.46·17-s + 6.71·19-s + 1.12·20-s − 1.53·22-s − 0.694·23-s + 5.41·25-s − 3.50·26-s − 0.226·28-s − 7.43·29-s − 1.36·31-s + 1.94·32-s + 3.78·34-s − 2.10·35-s − 4.45·37-s − 10.2·38-s + 8.17·40-s − 0.509·41-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 0.173·4-s + 1.44·5-s − 0.246·7-s + 0.895·8-s − 1.56·10-s + 0.301·11-s + 0.635·13-s + 0.267·14-s − 1.14·16-s − 0.598·17-s + 1.54·19-s + 0.250·20-s − 0.326·22-s − 0.144·23-s + 1.08·25-s − 0.688·26-s − 0.0428·28-s − 1.38·29-s − 0.245·31-s + 0.343·32-s + 0.648·34-s − 0.355·35-s − 0.732·37-s − 1.66·38-s + 1.29·40-s − 0.0796·41-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 | \( 1 + 1.53T + 2T^{2} \) |
| 5 | \( 1 - 3.22T + 5T^{2} \) |
| 7 | \( 1 + 0.652T + 7T^{2} \) |
| 13 | \( 1 - 2.29T + 13T^{2} \) |
| 17 | \( 1 + 2.46T + 17T^{2} \) |
| 19 | \( 1 - 6.71T + 19T^{2} \) |
| 23 | \( 1 + 0.694T + 23T^{2} \) |
| 29 | \( 1 + 7.43T + 29T^{2} \) |
| 31 | \( 1 + 1.36T + 31T^{2} \) |
| 37 | \( 1 + 4.45T + 37T^{2} \) |
| 41 | \( 1 + 0.509T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 7.71T + 47T^{2} \) |
| 53 | \( 1 - 6.30T + 53T^{2} \) |
| 59 | \( 1 - 0.758T + 59T^{2} \) |
| 61 | \( 1 - 1.75T + 61T^{2} \) |
| 67 | \( 1 + 5.31T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 18.3T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.46740245984603772390706764634, −6.96576831524867417182596953699, −6.16754693917329413847638657654, −5.51148871512038763719723859412, −4.85475261577138040613586137169, −3.80512840457395777442496361176, −2.91281991902851062606811944480, −1.70770655908513108872228473541, −1.44430210278974547181356112073, 0,
1.44430210278974547181356112073, 1.70770655908513108872228473541, 2.91281991902851062606811944480, 3.80512840457395777442496361176, 4.85475261577138040613586137169, 5.51148871512038763719723859412, 6.16754693917329413847638657654, 6.96576831524867417182596953699, 7.46740245984603772390706764634
