L(s) = 1 | + (4.5 − 2.59i)7-s + (2.5 − 4.33i)13-s + 5.19i·19-s + (−2.5 − 4.33i)25-s + (−9 − 5.19i)31-s + 11·37-s + (−9 + 5.19i)43-s + (10 − 17.3i)49-s + (0.5 + 0.866i)61-s + (13.5 + 7.79i)67-s + 7·73-s + (4.5 − 2.59i)79-s − 25.9i·91-s + (−9.5 − 16.4i)97-s + (13.5 + 7.79i)103-s + ⋯ |
L(s) = 1 | + (1.70 − 0.981i)7-s + (0.693 − 1.20i)13-s + 1.19i·19-s + (−0.5 − 0.866i)25-s + (−1.61 − 0.933i)31-s + 1.80·37-s + (−1.37 + 0.792i)43-s + (1.42 − 2.47i)49-s + (0.0640 + 0.110i)61-s + (1.64 + 0.952i)67-s + 0.819·73-s + (0.506 − 0.292i)79-s − 2.72i·91-s + (−0.964 − 1.67i)97-s + (1.33 + 0.767i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.013532749\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.013532749\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-4.5 + 2.59i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 5.19iT - 19T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (9 + 5.19i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 11T + 37T^{2} \) |
| 41 | \( 1 + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (9 - 5.19i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-13.5 - 7.79i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 + (-4.5 + 2.59i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (9.5 + 16.4i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.718007030682469486275479393183, −8.328925532816339426799889938216, −8.046828685041279380352808564356, −7.35954374425345910510747919885, −6.09660609737121073298803901806, −5.32245723708886382928064509824, −4.33669709400533156588235149354, −3.59655871859952174450881675602, −2.03663130615716882751725528435, −0.942113657085336780854598286456,
1.50711162235389149707276890328, 2.31679278661667540193135661397, 3.77494864241125368317108564764, 4.83306592173024571555445160727, 5.38647248708788311457938252948, 6.46487655691979086427647405746, 7.40306571360408387840554494653, 8.268224373205767301190406580411, 8.940951847667308329536116191992, 9.470570090356143421798942086014