| L(s) = 1 | + (0.842 − 3.14i)2-s + (−0.0602 − 0.104i)3-s + (−5.71 − 3.30i)4-s + (−0.378 + 0.101i)6-s + (−1.56 − 5.82i)7-s + (−5.98 + 5.98i)8-s + (4.49 − 7.78i)9-s + (−15.8 − 4.25i)11-s + 0.795i·12-s + (9.23 + 9.15i)13-s − 19.6·14-s + (0.583 + 1.01i)16-s + (8.70 + 5.02i)17-s + (−20.6 − 20.6i)18-s + (−15.0 + 4.02i)19-s + ⋯ |
| L(s) = 1 | + (0.421 − 1.57i)2-s + (−0.0200 − 0.0347i)3-s + (−1.42 − 0.825i)4-s + (−0.0631 + 0.0169i)6-s + (−0.222 − 0.832i)7-s + (−0.748 + 0.748i)8-s + (0.499 − 0.864i)9-s + (−1.44 − 0.386i)11-s + 0.0662i·12-s + (0.710 + 0.704i)13-s − 1.40·14-s + (0.0364 + 0.0631i)16-s + (0.512 + 0.295i)17-s + (−1.14 − 1.14i)18-s + (−0.790 + 0.211i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.504122 + 1.29495i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.504122 + 1.29495i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + (-9.23 - 9.15i)T \) |
| good | 2 | \( 1 + (-0.842 + 3.14i)T + (-3.46 - 2i)T^{2} \) |
| 3 | \( 1 + (0.0602 + 0.104i)T + (-4.5 + 7.79i)T^{2} \) |
| 7 | \( 1 + (1.56 + 5.82i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (15.8 + 4.25i)T + (104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (-8.70 - 5.02i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (15.0 - 4.02i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (20.1 - 11.6i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (4.19 + 7.27i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (3.53 + 3.53i)T + 961iT^{2} \) |
| 37 | \( 1 + (-15.5 - 4.15i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-5.36 + 20.0i)T + (-1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (39.9 + 23.0i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-48.8 + 48.8i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 - 82.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (5.93 + 22.1i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-56.7 + 98.2i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-7.89 + 29.4i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (91.3 - 24.4i)T + (4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (-66.0 + 66.0i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 87.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-61.9 - 61.9i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (44.9 + 12.0i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (100. - 27.0i)T + (8.14e3 - 4.70e3i)T^{2} \) |
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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
−10.73498938022622676593884443874, −10.28140092036601043510437706470, −9.403168641613571911825253765714, −8.150455241162093218703430066844, −6.85920230846546229976830300297, −5.53591528087584120437149329933, −4.10058819869001952169238960192, −3.54649404215524433731864647275, −2.02883741933229854068569404473, −0.56496483546603370778042822310,
2.52788333978363020895920787462, 4.32787338298605226287499398957, 5.34773683957623761688418759245, 5.93955055373771865378137963372, 7.20456629291578138777208642454, 7.993699083388270444723793612766, 8.640860219712145729469807179561, 10.03695769090581782823898323096, 10.87619687945614925683328640685, 12.42471273677415588199222782688
