L(s) = 1 | + (0.809 + 0.587i)4-s + (0.309 + 0.951i)5-s − 1.90i·7-s + (−0.309 − 0.951i)9-s + (−0.190 + 0.587i)11-s + (0.309 + 0.951i)16-s + (0.690 + 0.951i)17-s + (−0.809 + 0.587i)19-s + (−0.309 + 0.951i)20-s + (−1.80 − 0.587i)23-s + (−0.809 + 0.587i)25-s + (1.11 − 1.53i)28-s + (1.80 − 0.587i)35-s + (0.309 − 0.951i)36-s − 1.17i·43-s + (−0.5 + 0.363i)44-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)4-s + (0.309 + 0.951i)5-s − 1.90i·7-s + (−0.309 − 0.951i)9-s + (−0.190 + 0.587i)11-s + (0.309 + 0.951i)16-s + (0.690 + 0.951i)17-s + (−0.809 + 0.587i)19-s + (−0.309 + 0.951i)20-s + (−1.80 − 0.587i)23-s + (−0.809 + 0.587i)25-s + (1.11 − 1.53i)28-s + (1.80 − 0.587i)35-s + (0.309 − 0.951i)36-s − 1.17i·43-s + (−0.5 + 0.363i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.020540789\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.020540789\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.309 - 0.951i)T \) |
| 19 | \( 1 + (0.809 - 0.587i)T \) |
good | 2 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 3 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + 1.90iT - T^{2} \) |
| 11 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + 1.17iT - T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.309 + 0.951i)T^{2} \) |
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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
−11.08564531258956489912086284274, −10.34626425106951248417943597613, −9.962491290393719875652342363670, −8.220554886297140339835032960158, −7.51818624697035741913515575625, −6.67256833850442352570675537296, −6.10710462215372443126127097996, −4.00343695033569974392658940336, −3.53146896102969444768753435180, −1.96735852384102586528512784752,
1.93112197225081561036352910249, 2.74203902942505081087523692548, 4.89924737773433295499930340551, 5.64023156466830927648206487152, 6.12224877423187121168136150023, 7.75546896515520546100149609462, 8.534048310231528204789974487993, 9.382467312482019860847898996729, 10.23193411506540867269356057908, 11.47442423723914685018147300023