| L(s) = 1 | − 4.24i·5-s + 3·9-s + 1.41i·13-s + 8·17-s − 12.9·25-s − 9.89i·29-s − 7.07i·37-s − 8·41-s − 12.7i·45-s − 7·49-s + 7.07i·53-s − 1.41i·61-s + 6·65-s − 6·73-s + 9·81-s + ⋯ |
| L(s) = 1 | − 1.89i·5-s + 9-s + 0.392i·13-s + 1.94·17-s − 2.59·25-s − 1.83i·29-s − 1.16i·37-s − 1.24·41-s − 1.89i·45-s − 49-s + 0.971i·53-s − 0.181i·61-s + 0.744·65-s − 0.702·73-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.700193975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.700193975\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 + 4.24iT - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 1.41iT - 13T^{2} \) |
| 17 | \( 1 - 8T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 9.89iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7.07iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 7.07iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 1.41iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−9.675544006041973787915482133883, −9.001836917033812640004181442563, −8.023156964905480116536809795543, −7.53912032241304628657285262237, −6.14139566230196011664583160679, −5.28693725778176371962631409225, −4.51370664754349320144915149208, −3.69901364075716829924793442575, −1.83461442087961783779300951014, −0.853150689798580744507892687969,
1.62423628305960304378141745421, 3.11404081173263232023388348299, 3.51422708460452973604437909679, 5.00051465117045982215831085052, 6.06633879820410064608686194122, 6.93945212878735872905995115407, 7.41392790345235778503646695965, 8.289619812722943542371047406248, 9.778821991887811741180680707096, 10.13081465049832535067708805009
