L(s) = 1 | + (0.118 + 0.673i)3-s + (0.500 − 0.181i)9-s + (0.300 − 0.173i)11-s + (0.939 + 0.342i)17-s + (−0.642 + 0.766i)19-s + (0.173 − 0.984i)25-s + (0.524 + 0.907i)27-s + (0.152 + 0.181i)33-s + (−1.26 + 0.223i)41-s + (0.642 + 0.766i)43-s + (0.5 + 0.866i)49-s + (−0.118 + 0.673i)51-s + (−0.592 − 0.342i)57-s + (−1.85 − 0.673i)59-s + (1.20 − 0.439i)67-s + ⋯ |
L(s) = 1 | + (0.118 + 0.673i)3-s + (0.500 − 0.181i)9-s + (0.300 − 0.173i)11-s + (0.939 + 0.342i)17-s + (−0.642 + 0.766i)19-s + (0.173 − 0.984i)25-s + (0.524 + 0.907i)27-s + (0.152 + 0.181i)33-s + (−1.26 + 0.223i)41-s + (0.642 + 0.766i)43-s + (0.5 + 0.866i)49-s + (−0.118 + 0.673i)51-s + (−0.592 − 0.342i)57-s + (−1.85 − 0.673i)59-s + (1.20 − 0.439i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.182560871\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.182560871\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (0.642 - 0.766i)T \) |
good | 3 | \( 1 + (-0.118 - 0.673i)T + (-0.939 + 0.342i)T^{2} \) |
| 5 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.300 + 0.173i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (1.26 - 0.223i)T + (0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (-0.642 - 0.766i)T + (-0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (1.85 + 0.673i)T + (0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (-1.20 + 0.439i)T + (0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (1.32 + 0.766i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (1.70 + 0.300i)T + (0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (-0.439 + 1.20i)T + (-0.766 - 0.642i)T^{2} \) |
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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
−10.07708319696340407521860443919, −9.313566528786407954258062683547, −8.451976466288999413472719163458, −7.68111717452451771480385205943, −6.61309937859255469102360372299, −5.84616383067320958003788467591, −4.71466321575011437885619951323, −3.97372989114457644526825441913, −3.06482814512033183405177947461, −1.52592928334195601004677255390,
1.29478865325345599634721666383, 2.44055951308115548738215604526, 3.66132266914202766266300951460, 4.73655849017714614837210645442, 5.67190278255918978419477308886, 6.81453485639260019699595900529, 7.23633888599951650297752466058, 8.136948208526445834431309992274, 8.981334628341220712058825923034, 9.834016594714501912359294201761