L(s) = 1 | − 1.73i·2-s − 1.99·4-s + (0.5 − 0.866i)5-s + 1.73i·7-s + 1.73i·8-s − 9-s + (−1.49 − 0.866i)10-s + 2.99·14-s + 0.999·16-s + 1.73i·18-s + 19-s + (−0.999 + 1.73i)20-s + (−0.499 − 0.866i)25-s − 3.46i·28-s − 31-s + ⋯ |
L(s) = 1 | − 1.73i·2-s − 1.99·4-s + (0.5 − 0.866i)5-s + 1.73i·7-s + 1.73i·8-s − 9-s + (−1.49 − 0.866i)10-s + 2.99·14-s + 0.999·16-s + 1.73i·18-s + 19-s + (−0.999 + 1.73i)20-s + (−0.499 − 0.866i)25-s − 3.46i·28-s − 31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6078176341\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6078176341\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 + 1.73iT - T^{2} \) |
| 3 | \( 1 + T^{2} \) |
| 7 | \( 1 - 1.73iT - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 1.73iT - T^{2} \) |
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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
−12.46543236559967768776943439337, −11.95316490223993661237857601247, −11.15479282252388540651046991477, −9.749128852540363752407603347420, −9.069309539962566150968174134441, −8.406564653898943386903774874381, −5.79099571878796810866265396472, −4.99559637066076860419486785128, −3.14553519709855962013572144934, −1.98470118185236666196526716097,
3.57720643177974959481399999539, 5.14454408149836053986851032894, 6.28352037519677019263430211854, 7.16240109251553910414852154377, 7.87127156377874814137322110235, 9.234470675715962152120679315755, 10.30377698260116748695414024712, 11.32697481286746022943537966114, 13.24129811164260988978612131150, 14.01728269487878139040519980875