L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·8-s + (0.5 + 0.866i)9-s + (0.499 + 0.866i)10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (−1.5 + 0.866i)17-s − 0.999·18-s − 0.999·20-s + (−0.499 − 0.866i)25-s + (0.499 + 0.866i)26-s + (−0.5 + 0.866i)29-s + (−0.499 − 0.866i)32-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·8-s + (0.5 + 0.866i)9-s + (0.499 + 0.866i)10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (−1.5 + 0.866i)17-s − 0.999·18-s − 0.999·20-s + (−0.499 − 0.866i)25-s + (0.499 + 0.866i)26-s + (−0.5 + 0.866i)29-s + (−0.499 − 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6224374348\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6224374348\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.73iT - T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−12.70843432330213825977163120907, −10.92964267101808204889946191360, −10.32861644157476669695911839420, −9.153176252344279748857334196517, −8.473277589334956967593091506391, −7.52719041942000516695662327169, −6.27672462645822086311697588438, −5.31678348172953065925492238190, −4.33426405051252463644674277634, −1.75313437929249790249907423913,
1.96535227925586864006146446441, 3.33248629198086177599125501314, 4.53003450634788901396152684989, 6.42410185191117863987556631355, 7.14700182485008112197213593774, 8.606974884052520724801682866152, 9.506389038712524404579253548434, 10.13388902225517093990909170710, 11.36956363915845739728229405761, 11.66955307630825582851493408218