| L(s) = 1 | + (1.41 − 4.36i)2-s + (−10.5 − 7.69i)4-s + (−1.19 − 11.1i)5-s − 20.8·7-s + (−18.8 + 13.7i)8-s + (−50.2 − 10.5i)10-s + (2.21 − 6.80i)11-s + (9.36 + 28.8i)13-s + (−29.6 + 91.2i)14-s + (0.784 + 2.41i)16-s + (−22.6 + 16.4i)17-s + (−9.78 + 7.11i)19-s + (−72.8 + 126. i)20-s + (−26.5 − 19.3i)22-s + (16.1 − 49.6i)23-s + ⋯ |
| L(s) = 1 | + (0.501 − 1.54i)2-s + (−1.32 − 0.961i)4-s + (−0.106 − 0.994i)5-s − 1.12·7-s + (−0.834 + 0.606i)8-s + (−1.58 − 0.334i)10-s + (0.0606 − 0.186i)11-s + (0.199 + 0.614i)13-s + (−0.565 + 1.74i)14-s + (0.0122 + 0.0377i)16-s + (−0.323 + 0.235i)17-s + (−0.118 + 0.0858i)19-s + (−0.815 + 1.41i)20-s + (−0.257 − 0.187i)22-s + (0.146 − 0.450i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.149 - 0.988i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.149 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.641051 + 0.745611i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.641051 + 0.745611i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.19 + 11.1i)T \) |
| good | 2 | \( 1 + (-1.41 + 4.36i)T + (-6.47 - 4.70i)T^{2} \) |
| 7 | \( 1 + 20.8T + 343T^{2} \) |
| 11 | \( 1 + (-2.21 + 6.80i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (-9.36 - 28.8i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (22.6 - 16.4i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (9.78 - 7.11i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-16.1 + 49.6i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-119. - 86.6i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (172. - 125. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (22.0 + 67.7i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (159. + 489. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 176.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (181. + 132. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-292. - 212. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (181. + 558. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-244. + 752. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-347. + 252. i)T + (9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (321. + 233. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (321. - 988. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (989. + 718. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-189. + 137. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-56.9 + 175. i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (67.1 + 48.7i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−11.22176814260299994842644064334, −10.30016825154779226329027700341, −9.348250633406949147031311202228, −8.656378234972356487881644938004, −6.85045233285815532058996396594, −5.42779452006519169884048347450, −4.25868644551262816904027176624, −3.32626351035418610196468045465, −1.82019700436407885109058502451, −0.33575395575140502558398439773,
2.95665974177821613157934914168, 4.16408370788388201942824381996, 5.65989294482335775808079833623, 6.46818007222058771220587032240, 7.17907615453857364207689054824, 8.133879798241034699666685351734, 9.419261780186179377434109845573, 10.41931222667024792877736691785, 11.64718484689430301554835184052, 13.04806774280397152801933335910
