L(s) = 1 | + (0.5 − 0.866i)7-s + 2·13-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)31-s + (−1 − 1.73i)37-s + 43-s + (−0.499 − 0.866i)49-s + (0.5 + 0.866i)61-s + (1 − 1.73i)67-s + (0.5 − 0.866i)73-s + (1 + 1.73i)79-s + (1 − 1.73i)91-s − 97-s + (1 + 1.73i)103-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)7-s + 2·13-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)31-s + (−1 − 1.73i)37-s + 43-s + (−0.499 − 0.866i)49-s + (0.5 + 0.866i)61-s + (1 − 1.73i)67-s + (0.5 − 0.866i)73-s + (1 + 1.73i)79-s + (1 − 1.73i)91-s − 97-s + (1 + 1.73i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.403825573\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.403825573\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - 2T + T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−8.815271575623316302579779526192, −8.128001158619050544783132378622, −7.30878386537984907077780515697, −6.65823201422331666216501939357, −5.78913769048769684396343853399, −4.98411873408828019884347305389, −3.92651313435897278522217883641, −3.52997627047824460349164794222, −2.05408685603753019419363762605, −1.02668165045805675640069673409,
1.37554072832928014524620016654, 2.30514510687235254119250570635, 3.49107410634711396724391598675, 4.19258181092855002355422033845, 5.25370995451373661846185443757, 6.03357412953258267668983010155, 6.43994571219116060256033335697, 7.67341998901374022765975351397, 8.470877040860213199680862253172, 8.636490401739131474921612612021