L(s) = 1 | + (1.16 − 0.799i)2-s − 3-s + (0.720 − 1.86i)4-s − 0.864i·5-s + (−1.16 + 0.799i)6-s + 1.73·7-s + (−0.651 − 2.75i)8-s + 9-s + (−0.691 − 1.00i)10-s + 5.29·11-s + (−0.720 + 1.86i)12-s + 1.41i·13-s + (2.02 − 1.38i)14-s + 0.864i·15-s + (−2.96 − 2.68i)16-s − 4.82·17-s + ⋯ |
L(s) = 1 | + (0.824 − 0.565i)2-s − 0.577·3-s + (0.360 − 0.932i)4-s − 0.386i·5-s + (−0.476 + 0.326i)6-s + 0.655·7-s + (−0.230 − 0.973i)8-s + 0.333·9-s + (−0.218 − 0.318i)10-s + 1.59·11-s + (−0.207 + 0.538i)12-s + 0.391i·13-s + (0.540 − 0.370i)14-s + 0.223i·15-s + (−0.740 − 0.672i)16-s − 1.17·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48344 - 1.70877i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48344 - 1.70877i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.16 + 0.799i)T \) |
| 3 | \( 1 + T \) |
| 67 | \( 1 + (-7.97 + 1.84i)T \) |
good | 5 | \( 1 + 0.864iT - 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 - 5.29T + 11T^{2} \) |
| 13 | \( 1 - 1.41iT - 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 + 4.26iT - 19T^{2} \) |
| 23 | \( 1 + 1.92iT - 23T^{2} \) |
| 29 | \( 1 + 2.13T + 29T^{2} \) |
| 31 | \( 1 - 5.52T + 31T^{2} \) |
| 37 | \( 1 + 7.24T + 37T^{2} \) |
| 41 | \( 1 + 0.306iT - 41T^{2} \) |
| 43 | \( 1 + 1.13T + 43T^{2} \) |
| 47 | \( 1 + 11.9iT - 47T^{2} \) |
| 53 | \( 1 + 8.91iT - 53T^{2} \) |
| 59 | \( 1 - 14.4iT - 59T^{2} \) |
| 61 | \( 1 - 12.4iT - 61T^{2} \) |
| 71 | \( 1 - 8.13iT - 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 8.20T + 79T^{2} \) |
| 83 | \( 1 + 0.390iT - 83T^{2} \) |
| 89 | \( 1 + 5.73T + 89T^{2} \) |
| 97 | \( 1 - 14.5iT - 97T^{2} \) |
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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
−10.31081904196380194845766387339, −9.227832194934497443194578985064, −8.628161547252042375156803885523, −6.83530942688501912479376950456, −6.62949119712470806612215864464, −5.28425363114198328641599395851, −4.59494355108010983052381519442, −3.83002860709773391866718525971, −2.20263302654570733359063539438, −1.03113108402797166826363700924,
1.74813041968215313444118646087, 3.34828660525836886888802270379, 4.31613365926688369647743772239, 5.12501960089707910534259642686, 6.29160334635905830408023808367, 6.66246127488580625129864246392, 7.72271831623207778428937351015, 8.588709189087377204195953911870, 9.601135683231801959841150860126, 10.97098927610772630973593582873