L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (1.30 − 0.951i)13-s + (−0.809 + 0.587i)16-s + (0.190 − 0.587i)17-s + 18-s − 1.61·26-s + (0.190 + 0.587i)29-s + 32-s + (−0.5 + 0.363i)34-s + (−0.809 − 0.587i)36-s + (−0.5 + 0.363i)37-s + (1.30 − 0.951i)41-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (1.30 − 0.951i)13-s + (−0.809 + 0.587i)16-s + (0.190 − 0.587i)17-s + 18-s − 1.61·26-s + (0.190 + 0.587i)29-s + 32-s + (−0.5 + 0.363i)34-s + (−0.809 − 0.587i)36-s + (−0.5 + 0.363i)37-s + (1.30 − 0.951i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8043669551\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8043669551\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−8.836794948844011131875188692137, −8.459485724253044539205796933030, −7.73799725874098532654204772274, −6.93156540062914761065020880410, −5.92656706472065396111437987045, −5.14382458355951634825515537200, −3.86946682190427220348141451577, −3.11328672026137483943424580955, −2.23444745983071046506044731110, −0.900456715950755990029111354991,
1.06815228181934673297886953534, 2.25898291840861027747833050121, 3.52906757319584919597598803973, 4.50679504468833932241103769960, 5.73440878559893036806229518181, 6.15227828822242522152735180767, 6.84163730965286167912515110580, 7.83772600348834377436165991896, 8.488103395688779964656832981053, 9.082144407673108099980334059526