L(s) = 1 | − 1.88·2-s + 2.22·3-s + 1.54·4-s + 0.640·5-s − 4.19·6-s − 4.86·7-s + 0.850·8-s + 1.96·9-s − 1.20·10-s + 4.50·11-s + 3.45·12-s − 0.00706·13-s + 9.16·14-s + 1.42·15-s − 4.69·16-s − 6.43·17-s − 3.70·18-s − 19-s + 0.992·20-s − 10.8·21-s − 8.49·22-s − 0.234·23-s + 1.89·24-s − 4.58·25-s + 0.0133·26-s − 2.30·27-s − 7.53·28-s + ⋯ |
L(s) = 1 | − 1.33·2-s + 1.28·3-s + 0.774·4-s + 0.286·5-s − 1.71·6-s − 1.83·7-s + 0.300·8-s + 0.655·9-s − 0.381·10-s + 1.35·11-s + 0.995·12-s − 0.00195·13-s + 2.44·14-s + 0.368·15-s − 1.17·16-s − 1.56·17-s − 0.872·18-s − 0.229·19-s + 0.221·20-s − 2.36·21-s − 1.81·22-s − 0.0488·23-s + 0.387·24-s − 0.917·25-s + 0.00260·26-s − 0.443·27-s − 1.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 + T \) |
| 317 | \( 1 + T \) |
good | 2 | \( 1 + 1.88T + 2T^{2} \) |
| 3 | \( 1 - 2.22T + 3T^{2} \) |
| 5 | \( 1 - 0.640T + 5T^{2} \) |
| 7 | \( 1 + 4.86T + 7T^{2} \) |
| 11 | \( 1 - 4.50T + 11T^{2} \) |
| 13 | \( 1 + 0.00706T + 13T^{2} \) |
| 17 | \( 1 + 6.43T + 17T^{2} \) |
| 23 | \( 1 + 0.234T + 23T^{2} \) |
| 29 | \( 1 - 8.13T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 - 1.87T + 37T^{2} \) |
| 41 | \( 1 - 0.709T + 41T^{2} \) |
| 43 | \( 1 - 2.98T + 43T^{2} \) |
| 47 | \( 1 + 8.81T + 47T^{2} \) |
| 53 | \( 1 - 6.85T + 53T^{2} \) |
| 59 | \( 1 + 9.51T + 59T^{2} \) |
| 61 | \( 1 - 2.64T + 61T^{2} \) |
| 67 | \( 1 - 5.68T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 3.68T + 73T^{2} \) |
| 79 | \( 1 + 9.08T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 9.09T + 89T^{2} \) |
| 97 | \( 1 + 7.06T + 97T^{2} \) |
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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
−8.098452916113819359569383313217, −7.05135810050656989016470048110, −6.58671922613647329679862877197, −6.12590168655711934484766640267, −4.44662674157611688536597451235, −3.90706169784801961478864918825, −2.87445147896953875772996176796, −2.35966408425215181696714464887, −1.24445322398853853526178090195, 0,
1.24445322398853853526178090195, 2.35966408425215181696714464887, 2.87445147896953875772996176796, 3.90706169784801961478864918825, 4.44662674157611688536597451235, 6.12590168655711934484766640267, 6.58671922613647329679862877197, 7.05135810050656989016470048110, 8.098452916113819359569383313217