L(s) = 1 | + (−0.416 + 1.35i)2-s + (0.966 + 1.43i)3-s + (−1.65 − 1.12i)4-s + (−1.57 − 1.57i)5-s + (−2.34 + 0.708i)6-s + 2.24·7-s + (2.20 − 1.76i)8-s + (−1.13 + 2.77i)9-s + (2.77 − 1.47i)10-s + (1.13 − 1.13i)11-s + (0.0176 − 3.46i)12-s + (−3.24 − 3.24i)13-s + (−0.935 + 3.04i)14-s + (0.739 − 3.77i)15-s + (1.47 + 3.71i)16-s − 1.66i·17-s + ⋯ |
L(s) = 1 | + (−0.294 + 0.955i)2-s + (0.558 + 0.829i)3-s + (−0.826 − 0.562i)4-s + (−0.702 − 0.702i)5-s + (−0.957 + 0.289i)6-s + 0.850·7-s + (0.780 − 0.624i)8-s + (−0.377 + 0.926i)9-s + (0.878 − 0.465i)10-s + (0.341 − 0.341i)11-s + (0.00510 − 0.999i)12-s + (−0.901 − 0.901i)13-s + (−0.250 + 0.812i)14-s + (0.191 − 0.975i)15-s + (0.367 + 0.929i)16-s − 0.403i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.582356 + 0.471527i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.582356 + 0.471527i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.416 - 1.35i)T \) |
| 3 | \( 1 + (-0.966 - 1.43i)T \) |
good | 5 | \( 1 + (1.57 + 1.57i)T + 5iT^{2} \) |
| 7 | \( 1 - 2.24T + 7T^{2} \) |
| 11 | \( 1 + (-1.13 + 1.13i)T - 11iT^{2} \) |
| 13 | \( 1 + (3.24 + 3.24i)T + 13iT^{2} \) |
| 17 | \( 1 + 1.66iT - 17T^{2} \) |
| 19 | \( 1 + (3.77 - 3.77i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.26iT - 23T^{2} \) |
| 29 | \( 1 + (3.23 - 3.23i)T - 29iT^{2} \) |
| 31 | \( 1 - 1.30iT - 31T^{2} \) |
| 37 | \( 1 + (-2.30 + 2.30i)T - 37iT^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + (-3.77 - 3.77i)T + 43iT^{2} \) |
| 47 | \( 1 + 3.74T + 47T^{2} \) |
| 53 | \( 1 + (-0.972 - 0.972i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3.88 + 3.88i)T - 59iT^{2} \) |
| 61 | \( 1 + (-4.19 - 4.19i)T + 61iT^{2} \) |
| 67 | \( 1 + (-8.02 + 8.02i)T - 67iT^{2} \) |
| 71 | \( 1 - 11.0iT - 71T^{2} \) |
| 73 | \( 1 + 6.38iT - 73T^{2} \) |
| 79 | \( 1 - 2.69iT - 79T^{2} \) |
| 83 | \( 1 + (-2.61 - 2.61i)T + 83iT^{2} \) |
| 89 | \( 1 + 7.35T + 89T^{2} \) |
| 97 | \( 1 + 5.67T + 97T^{2} \) |
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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
−15.95765627358636941087183709555, −14.88814190478150671465303862616, −14.27698974318256677014167699165, −12.71094722110332584301234446720, −10.98694456610594692203020077612, −9.620381904934211435696605197647, −8.425524526747607914988542948331, −7.72003282548528716475213835992, −5.36513628414595212095012969214, −4.21289449256991863971045032813,
2.29677960166614722649074286943, 4.19823136021397385132735328492, 7.06333713343041388551681232197, 8.118671092991725557645781522504, 9.370332564151671414979017907045, 11.06767496985668599272580490250, 11.83828098129248660494065931694, 12.94351548512331806788677022497, 14.31039607800830541924974555226, 14.93386674333398306069579542582