L(s) = 1 | + (−0.331 − 0.573i)3-s + (1.49 − 2.59i)5-s + (2.42 + 1.06i)7-s + (1.28 − 2.21i)9-s + (2.82 + 4.89i)11-s − 1.87·13-s − 1.98·15-s + (0.0960 + 0.166i)17-s + (0.986 − 1.70i)19-s + (−0.189 − 1.74i)21-s + (0.5 − 0.866i)23-s + (−1.97 − 3.42i)25-s − 3.68·27-s − 0.907·29-s + (−1.91 − 3.31i)31-s + ⋯ |
L(s) = 1 | + (−0.191 − 0.331i)3-s + (0.669 − 1.15i)5-s + (0.915 + 0.403i)7-s + (0.426 − 0.739i)9-s + (0.852 + 1.47i)11-s − 0.521·13-s − 0.511·15-s + (0.0232 + 0.0403i)17-s + (0.226 − 0.391i)19-s + (−0.0414 − 0.380i)21-s + (0.104 − 0.180i)23-s + (−0.395 − 0.684i)25-s − 0.708·27-s − 0.168·29-s + (−0.343 − 0.594i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.63962 - 0.767104i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.63962 - 0.767104i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.42 - 1.06i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
good | 3 | \( 1 + (0.331 + 0.573i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.49 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.82 - 4.89i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.87T + 13T^{2} \) |
| 17 | \( 1 + (-0.0960 - 0.166i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.986 + 1.70i)T + (-9.5 - 16.4i)T^{2} \) |
| 29 | \( 1 + 0.907T + 29T^{2} \) |
| 31 | \( 1 + (1.91 + 3.31i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.941 + 1.63i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6.12T + 41T^{2} \) |
| 43 | \( 1 - 9.84T + 43T^{2} \) |
| 47 | \( 1 + (-2.08 + 3.61i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.178 + 0.310i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.10 + 8.83i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.46 - 9.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.52 - 9.57i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 + (3.22 + 5.58i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.23 + 3.86i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.10T + 83T^{2} \) |
| 89 | \( 1 + (0.163 - 0.282i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 1.09T + 97T^{2} \) |
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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
−10.27517547108042065951279325472, −9.263071305459250938087005032638, −9.088661267660301739924343270629, −7.73668319693025820300483642457, −6.93626163158519626116527386223, −5.82433549847124267409229610658, −4.89923876619397457398217202844, −4.18130475220740173534537731929, −2.12127766552681894390218266731, −1.23449456237805614134743020980,
1.59009471461584419158468951490, 2.96145434205543956216973939723, 4.12522162541071698650075725691, 5.28764119651749127417160112593, 6.15279295410392630830924727259, 7.14933616796633004636988891752, 7.949709259387656131149732029296, 9.059971845138371012419149416278, 10.04499244300017676326043222294, 10.81561957393005272160863170127