L(s) = 1 | + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.951 + 0.309i)7-s + (0.587 − 0.809i)8-s + (−0.587 − 0.809i)9-s + (−0.587 + 0.809i)11-s + 0.999·14-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)18-s + (−0.309 + 0.951i)22-s + 0.618·23-s + (−0.951 + 0.309i)25-s + (0.951 − 0.309i)28-s + (−0.278 + 0.142i)29-s − i·32-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.951 + 0.309i)7-s + (0.587 − 0.809i)8-s + (−0.587 − 0.809i)9-s + (−0.587 + 0.809i)11-s + 0.999·14-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)18-s + (−0.309 + 0.951i)22-s + 0.618·23-s + (−0.951 + 0.309i)25-s + (0.951 − 0.309i)28-s + (−0.278 + 0.142i)29-s − i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.923243462\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.923243462\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 7 | \( 1 + (-0.951 - 0.309i)T \) |
| 11 | \( 1 + (0.587 - 0.809i)T \) |
good | 3 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 5 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 13 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 17 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 23 | \( 1 - 0.618T + T^{2} \) |
| 29 | \( 1 + (0.278 - 0.142i)T + (0.587 - 0.809i)T^{2} \) |
| 31 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.412i)T + (0.587 - 0.809i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + (-0.221 - 0.221i)T + iT^{2} \) |
| 47 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (1.95 + 0.309i)T + (0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 61 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 67 | \( 1 + (-1.39 - 1.39i)T + iT^{2} \) |
| 71 | \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−9.948671575745591670007337313524, −9.148536527449133782988559262567, −8.072301855406224916326389503585, −7.26317947542871418776041979331, −6.30979088552893573081478661627, −5.40988506057075849230029476274, −4.78589431303318741843819182052, −3.73177922677112751403700868077, −2.68205690368469831050429699293, −1.61603077252412623359653102771,
1.91380894841679167203602483109, 2.95737258679542689595701699079, 4.04950782119140188580294417404, 5.07365000622849027607274593773, 5.52794249197787132271337127367, 6.54824214676205771871403585133, 7.72379756410877641561140303842, 7.986181478786767631227453049606, 8.904126426923552338724722002421, 10.34502186583521205621417294898