| L(s) = 1 | + (2.5 + 4.33i)5-s + (−0.606 + 1.05i)7-s + (16.8 − 29.1i)11-s + (−38.7 − 67.1i)13-s − 73.2·17-s + 39.2·19-s + (17.6 + 30.4i)23-s + (−12.5 + 21.6i)25-s + (−7.89 + 13.6i)29-s + (−26.1 − 45.3i)31-s − 6.06·35-s − 67.1·37-s + (−12.5 − 21.6i)41-s + (15.8 − 27.4i)43-s + (−206. + 357. i)47-s + ⋯ |
| L(s) = 1 | + (0.223 + 0.387i)5-s + (−0.0327 + 0.0567i)7-s + (0.460 − 0.797i)11-s + (−0.827 − 1.43i)13-s − 1.04·17-s + 0.473·19-s + (0.159 + 0.276i)23-s + (−0.100 + 0.173i)25-s + (−0.0505 + 0.0875i)29-s + (−0.151 − 0.262i)31-s − 0.0292·35-s − 0.298·37-s + (−0.0476 − 0.0826i)41-s + (0.0562 − 0.0974i)43-s + (−0.639 + 1.10i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4763641582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4763641582\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.5 - 4.33i)T \) |
| good | 7 | \( 1 + (0.606 - 1.05i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-16.8 + 29.1i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (38.7 + 67.1i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 73.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-17.6 - 30.4i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (7.89 - 13.6i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (26.1 + 45.3i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 67.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + (12.5 + 21.6i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-15.8 + 27.4i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (206. - 357. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 54.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-257. - 446. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (431. - 747. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-183. - 318. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 930.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 797.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-42.7 + 74.1i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-45.9 + 79.6i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 888.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (166. - 288. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−9.337505712809945124791265041789, −8.584565533538289744220670185999, −7.69268006374772660420268560827, −6.99277604043825281322079836923, −6.01355255048029109764222282347, −5.40588161209388347865926847675, −4.32709546181519320358296256189, −3.21218062919657713522452473773, −2.52612500074183597435230067988, −1.11651925315463418446353993631,
0.10488064277868972119346716566, 1.62695826077331909101729137885, 2.32889423483688059881804612852, 3.77312880881898991305815714749, 4.61356757903997577749432992305, 5.22311926180340597453700883777, 6.64152127004046244278753708678, 6.84353664300222480687692836237, 7.940881765423081235410502771755, 8.932548217894031055289644015854
