L(s) = 1 | + (−1.41 − 0.0966i)2-s − 3-s + (1.98 + 0.272i)4-s − 2.83i·5-s + (1.41 + 0.0966i)6-s + 0.830·7-s + (−2.76 − 0.576i)8-s + 9-s + (−0.274 + 4.00i)10-s + 3.46·11-s + (−1.98 − 0.272i)12-s − 0.104i·13-s + (−1.17 − 0.0802i)14-s + 2.83i·15-s + (3.85 + 1.08i)16-s + 4.98·17-s + ⋯ |
L(s) = 1 | + (−0.997 − 0.0683i)2-s − 0.577·3-s + (0.990 + 0.136i)4-s − 1.27i·5-s + (0.576 + 0.0394i)6-s + 0.313·7-s + (−0.979 − 0.203i)8-s + 0.333·9-s + (−0.0867 + 1.26i)10-s + 1.04·11-s + (−0.571 − 0.0786i)12-s − 0.0289i·13-s + (−0.313 − 0.0214i)14-s + 0.733i·15-s + (0.962 + 0.270i)16-s + 1.20·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.737683 - 0.515097i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.737683 - 0.515097i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 + 0.0966i)T \) |
| 3 | \( 1 + T \) |
| 67 | \( 1 + (3.84 + 7.22i)T \) |
good | 5 | \( 1 + 2.83iT - 5T^{2} \) |
| 7 | \( 1 - 0.830T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 0.104iT - 13T^{2} \) |
| 17 | \( 1 - 4.98T + 17T^{2} \) |
| 19 | \( 1 - 5.61iT - 19T^{2} \) |
| 23 | \( 1 + 7.10iT - 23T^{2} \) |
| 29 | \( 1 + 0.746T + 29T^{2} \) |
| 31 | \( 1 + 2.40T + 31T^{2} \) |
| 37 | \( 1 - 0.364T + 37T^{2} \) |
| 41 | \( 1 - 9.73iT - 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 3.75iT - 47T^{2} \) |
| 53 | \( 1 + 6.44iT - 53T^{2} \) |
| 59 | \( 1 + 14.0iT - 59T^{2} \) |
| 61 | \( 1 + 8.51iT - 61T^{2} \) |
| 71 | \( 1 - 6.72iT - 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 2.89T + 79T^{2} \) |
| 83 | \( 1 + 12.2iT - 83T^{2} \) |
| 89 | \( 1 - 0.183T + 89T^{2} \) |
| 97 | \( 1 - 11.8iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.896644592459021819160442349603, −9.346263934069473532584722455492, −8.357637084702702344117121346637, −7.86850051676886991932566551128, −6.61825188518670889132925713194, −5.83798639853933309850643013347, −4.80626709664978177729032334567, −3.61732442070147101405330974461, −1.74914349830556242298361544458, −0.814099668160534009859036020399,
1.23687518462739566124161942694, 2.67786349212123299718818942914, 3.80442237219712283161322255132, 5.48203830236059491692724162769, 6.25993147604829690596686043081, 7.23931270224516137256175895244, 7.49834201872637867467757556607, 8.945031269966744139551090198192, 9.588730207756945340805237496094, 10.51459959513849805983010033335