L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + 0.999·6-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.866 + 0.499i)12-s + (−0.5 − 0.866i)16-s + i·17-s + (−0.866 − 0.499i)18-s − 2·19-s + (−0.866 − 0.499i)22-s + (0.499 − 0.866i)24-s − 0.999i·27-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + 0.999·6-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.866 + 0.499i)12-s + (−0.5 − 0.866i)16-s + i·17-s + (−0.866 − 0.499i)18-s − 2·19-s + (−0.866 − 0.499i)22-s + (0.499 − 0.866i)24-s − 0.999i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3983499006\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3983499006\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - iT - T^{2} \) |
| 19 | \( 1 + 2T + T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 2iT - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
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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
−9.911528475078839110408799632546, −8.637867402844645434691265742481, −8.199883273305099593035734116704, −7.17673155976640400362408448939, −6.56875302196552206070886735752, −6.06203510735435564077399046923, −5.01416789179383666717981765888, −4.17447292961991938419969427431, −2.25859564692299210050947729146, −1.42860540125976050099974465698,
0.45395563423841967327361832375, 1.97058776441899405761209405845, 3.32332478407927401159022863107, 4.10187096955129550011389501728, 5.12881101680909443395623677804, 6.30625675643013660697211699987, 6.72438027726173487667877908285, 7.79556411831064873483009761563, 8.832899843128343455092852818108, 9.147237538689713614236337955875