L(s) = 1 | + (−0.992 − 0.125i)2-s + (0.216 − 0.527i)3-s + (0.968 + 0.248i)4-s + (−0.280 + 0.496i)6-s + (−0.929 − 0.368i)8-s + (0.480 + 0.474i)9-s + (0.891 + 0.302i)11-s + (0.340 − 0.456i)12-s + (0.876 + 0.481i)16-s + (−0.562 − 0.256i)17-s + (−0.417 − 0.530i)18-s + (0.635 + 0.567i)19-s + (−0.846 − 0.411i)22-s + (−0.395 + 0.410i)24-s + (−0.425 + 0.904i)25-s + ⋯ |
L(s) = 1 | + (−0.992 − 0.125i)2-s + (0.216 − 0.527i)3-s + (0.968 + 0.248i)4-s + (−0.280 + 0.496i)6-s + (−0.929 − 0.368i)8-s + (0.480 + 0.474i)9-s + (0.891 + 0.302i)11-s + (0.340 − 0.456i)12-s + (0.876 + 0.481i)16-s + (−0.562 − 0.256i)17-s + (−0.417 − 0.530i)18-s + (0.635 + 0.567i)19-s + (−0.846 − 0.411i)22-s + (−0.395 + 0.410i)24-s + (−0.425 + 0.904i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8957651892\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8957651892\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.992 + 0.125i)T \) |
| 251 | \( 1 + (0.514 + 0.857i)T \) |
good | 3 | \( 1 + (-0.216 + 0.527i)T + (-0.711 - 0.702i)T^{2} \) |
| 5 | \( 1 + (0.425 - 0.904i)T^{2} \) |
| 7 | \( 1 + (0.470 + 0.882i)T^{2} \) |
| 11 | \( 1 + (-0.891 - 0.302i)T + (0.793 + 0.607i)T^{2} \) |
| 13 | \( 1 + (-0.577 + 0.816i)T^{2} \) |
| 17 | \( 1 + (0.562 + 0.256i)T + (0.656 + 0.754i)T^{2} \) |
| 19 | \( 1 + (-0.635 - 0.567i)T + (0.112 + 0.993i)T^{2} \) |
| 23 | \( 1 + (-0.920 + 0.391i)T^{2} \) |
| 29 | \( 1 + (-0.762 - 0.647i)T^{2} \) |
| 31 | \( 1 + (0.999 + 0.0251i)T^{2} \) |
| 37 | \( 1 + (0.947 - 0.320i)T^{2} \) |
| 41 | \( 1 + (0.618 - 1.15i)T + (-0.556 - 0.830i)T^{2} \) |
| 43 | \( 1 + (-1.35 - 0.0339i)T + (0.998 + 0.0502i)T^{2} \) |
| 47 | \( 1 + (-0.535 + 0.844i)T^{2} \) |
| 53 | \( 1 + (0.778 - 0.627i)T^{2} \) |
| 59 | \( 1 + (0.0228 - 0.0104i)T + (0.656 - 0.754i)T^{2} \) |
| 61 | \( 1 + (0.837 + 0.546i)T^{2} \) |
| 67 | \( 1 + (-1.70 - 0.302i)T + (0.938 + 0.344i)T^{2} \) |
| 71 | \( 1 + (0.597 + 0.801i)T^{2} \) |
| 73 | \( 1 + (-0.0886 + 0.364i)T + (-0.888 - 0.459i)T^{2} \) |
| 79 | \( 1 + (-0.850 - 0.525i)T^{2} \) |
| 83 | \( 1 + (0.520 + 1.92i)T + (-0.863 + 0.503i)T^{2} \) |
| 89 | \( 1 + (0.00850 - 0.0748i)T + (-0.974 - 0.224i)T^{2} \) |
| 97 | \( 1 + (1.54 - 1.00i)T + (0.402 - 0.915i)T^{2} \) |
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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
−9.416988540477123716385768420408, −8.489357211908860666205176932566, −7.82097313950160717924079105976, −7.11348041647228111367522898932, −6.57994779784434160625208936210, −5.53663222025366334622129696738, −4.29498215043917823771810075025, −3.22041992083334381507278666298, −2.07423507575622757258591333731, −1.29433988071119364317449777710,
1.05162899852608988171362773779, 2.35809418599449773469491740432, 3.49298061301716189997685377953, 4.32922360519393691396774731732, 5.54201506707945691714086910648, 6.50070118079256123014133170297, 6.98918875963823852856582067121, 7.984221836819412899624658986210, 8.787541616397058412230866464279, 9.328160187452284515591786680506