L(s) = 1 | + 3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + 9-s + (−0.5 − 0.866i)15-s + (0.5 − 0.866i)21-s + (−1.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + 27-s − 1.73i·29-s − 0.999·35-s + 41-s − 43-s + (−0.5 − 0.866i)45-s + (1 + 1.73i)47-s + ⋯ |
L(s) = 1 | + 3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + 9-s + (−0.5 − 0.866i)15-s + (0.5 − 0.866i)21-s + (−1.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + 27-s − 1.73i·29-s − 0.999·35-s + 41-s − 43-s + (−0.5 − 0.866i)45-s + (1 + 1.73i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.520962514\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.520962514\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + 1.73iT - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.454773608195679887008379194942, −8.470009107532137536614341924555, −7.83310387459976979389831335563, −7.52274560645316690753108591571, −6.28725570900515319270898399069, −5.10049163080790931526605806148, −4.12751978795193242312977588963, −3.80711017784636327944538132443, −2.32357199325351873157807226934, −1.17896150314427398360588005636,
1.90310098412883740314056132220, 2.70631165555979516421769316171, 3.62454264742552273014860565001, 4.48920987627078146254991613648, 5.62703403777014058933266887230, 6.68031155306567719063254269574, 7.35719085514996154521453302151, 8.276129726807801814644861045428, 8.604610452302635385444119066389, 9.607458509375348379568536322599