L(s) = 1 | + (0.728 + 0.684i)2-s + (0.0627 + 0.998i)4-s + (0.781 − 0.733i)5-s + (−0.637 + 0.770i)8-s + (−0.929 − 0.368i)9-s + 1.07·10-s + (−0.328 + 0.180i)13-s + (−0.992 + 0.125i)16-s + (−0.5 − 0.363i)17-s + (−0.425 − 0.904i)18-s + (0.781 + 0.733i)20-s + (0.00932 − 0.148i)25-s + (−0.362 − 0.0931i)26-s + (−1.41 + 0.779i)29-s + (−0.809 − 0.587i)32-s + ⋯ |
L(s) = 1 | + (0.728 + 0.684i)2-s + (0.0627 + 0.998i)4-s + (0.781 − 0.733i)5-s + (−0.637 + 0.770i)8-s + (−0.929 − 0.368i)9-s + 1.07·10-s + (−0.328 + 0.180i)13-s + (−0.992 + 0.125i)16-s + (−0.5 − 0.363i)17-s + (−0.425 − 0.904i)18-s + (0.781 + 0.733i)20-s + (0.00932 − 0.148i)25-s + (−0.362 − 0.0931i)26-s + (−1.41 + 0.779i)29-s + (−0.809 − 0.587i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.207470862\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.207470862\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.728 - 0.684i)T \) |
| 101 | \( 1 + (-0.0627 - 0.998i)T \) |
good | 3 | \( 1 + (0.929 + 0.368i)T^{2} \) |
| 5 | \( 1 + (-0.781 + 0.733i)T + (0.0627 - 0.998i)T^{2} \) |
| 7 | \( 1 + (-0.535 - 0.844i)T^{2} \) |
| 11 | \( 1 + (-0.728 - 0.684i)T^{2} \) |
| 13 | \( 1 + (0.328 - 0.180i)T + (0.535 - 0.844i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.968 - 0.248i)T^{2} \) |
| 23 | \( 1 + (0.637 + 0.770i)T^{2} \) |
| 29 | \( 1 + (1.41 - 0.779i)T + (0.535 - 0.844i)T^{2} \) |
| 31 | \( 1 + (-0.535 - 0.844i)T^{2} \) |
| 37 | \( 1 + (0.0235 + 0.123i)T + (-0.929 + 0.368i)T^{2} \) |
| 41 | \( 1 + (-1.60 + 1.16i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + (-0.876 - 0.481i)T^{2} \) |
| 47 | \( 1 + (-0.876 + 0.481i)T^{2} \) |
| 53 | \( 1 + (-0.110 + 1.74i)T + (-0.992 - 0.125i)T^{2} \) |
| 59 | \( 1 + (-0.968 + 0.248i)T^{2} \) |
| 61 | \( 1 + (0.0800 + 1.27i)T + (-0.992 + 0.125i)T^{2} \) |
| 67 | \( 1 + (0.929 - 0.368i)T^{2} \) |
| 71 | \( 1 + (0.929 + 0.368i)T^{2} \) |
| 73 | \( 1 + (0.824 - 1.75i)T + (-0.637 - 0.770i)T^{2} \) |
| 79 | \( 1 + (0.637 - 0.770i)T^{2} \) |
| 83 | \( 1 + (0.637 - 0.770i)T^{2} \) |
| 89 | \( 1 + (-1.84 - 0.233i)T + (0.968 + 0.248i)T^{2} \) |
| 97 | \( 1 + (-0.110 - 1.74i)T + (-0.992 + 0.125i)T^{2} \) |
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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
−11.80604605846612985219955853762, −10.93188686156957912538798385189, −9.316743305182769298886486887278, −8.965504798062210882030332764742, −7.79620579704446194804503599628, −6.71346295789394127659764901614, −5.69180833476671623880333783064, −5.09665310514972593378497140383, −3.77571653576647881529836967252, −2.35699536337165054344293693292,
2.12865788153770767008274315388, 2.98532921749416349024449830239, 4.38921035350799099937578120352, 5.68831225960400433647677791144, 6.18934430949641370341460472237, 7.48521595213095589026946134790, 8.912070611387996128729526525677, 9.818955400658309282463512108626, 10.67110732039080522696554961294, 11.25749209511203387206908600300