| L(s) = 1 | + 2.80i·2-s − 0.554i·3-s − 5.85·4-s + 1.55·6-s − 3.04i·7-s − 10.7i·8-s + 2.69·9-s − 2.93·11-s + 3.24i·12-s − 3.24i·13-s + 8.54·14-s + 18.5·16-s − 2.15i·17-s + 7.54i·18-s + 19-s + ⋯ |
| L(s) = 1 | + 1.98i·2-s − 0.320i·3-s − 2.92·4-s + 0.634·6-s − 1.15i·7-s − 3.81i·8-s + 0.897·9-s − 0.886·11-s + 0.937i·12-s − 0.900i·13-s + 2.28·14-s + 4.63·16-s − 0.523i·17-s + 1.77i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.928330 + 0.219149i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.928330 + 0.219149i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.80iT - 2T^{2} \) |
| 3 | \( 1 + 0.554iT - 3T^{2} \) |
| 7 | \( 1 + 3.04iT - 7T^{2} \) |
| 11 | \( 1 + 2.93T + 11T^{2} \) |
| 13 | \( 1 + 3.24iT - 13T^{2} \) |
| 17 | \( 1 + 2.15iT - 17T^{2} \) |
| 23 | \( 1 + 1.19iT - 23T^{2} \) |
| 29 | \( 1 - 1.77T + 29T^{2} \) |
| 31 | \( 1 + 9.34T + 31T^{2} \) |
| 37 | \( 1 + 1.15iT - 37T^{2} \) |
| 41 | \( 1 - 8.57T + 41T^{2} \) |
| 43 | \( 1 + 5.27iT - 43T^{2} \) |
| 47 | \( 1 - 2.35iT - 47T^{2} \) |
| 53 | \( 1 + 8.82iT - 53T^{2} \) |
| 59 | \( 1 - 5.70T + 59T^{2} \) |
| 61 | \( 1 + 9.96T + 61T^{2} \) |
| 67 | \( 1 - 4.98iT - 67T^{2} \) |
| 71 | \( 1 - 2.70T + 71T^{2} \) |
| 73 | \( 1 + 13.7iT - 73T^{2} \) |
| 79 | \( 1 + 5.66T + 79T^{2} \) |
| 83 | \( 1 - 3.00iT - 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 3.24iT - 97T^{2} \) |
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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
−10.61592045633528684949453945570, −9.980031870470896089421101650175, −8.981239391301163627550663966713, −7.69419758391912148107997437293, −7.57497732345294696120004535289, −6.69127412730484052769146063552, −5.57574773541202239567828154445, −4.70248937170534383877256010340, −3.67975427645049979454676207583, −0.63230589045254169891980932520,
1.66574271046411139001224410892, 2.67238391842089591536089258971, 3.87284062161553547902962259639, 4.80060297209746597936582554436, 5.73176859208518170009877720723, 7.70688154620317537695053180861, 8.866119038112904528944402426121, 9.364791529799236567533219988804, 10.21062241042963116016371858346, 10.96651566917551750844933541280
