L(s) = 1 | + (−2.34 + 3.24i)2-s + (−4.63 + 4.63i)3-s + (−5.00 − 15.1i)4-s + (−29.2 + 29.2i)5-s + (−4.15 − 25.8i)6-s + 59.6·7-s + (60.9 + 19.3i)8-s + 38.0i·9-s + (−26.1 − 163. i)10-s + (−18.0 − 18.0i)11-s + (93.6 + 47.2i)12-s + (50.7 + 50.7i)13-s + (−139. + 193. i)14-s − 270. i·15-s + (−205. + 152. i)16-s − 223.·17-s + ⋯ |
L(s) = 1 | + (−0.586 + 0.810i)2-s + (−0.515 + 0.515i)3-s + (−0.313 − 0.949i)4-s + (−1.16 + 1.16i)5-s + (−0.115 − 0.719i)6-s + 1.21·7-s + (0.953 + 0.302i)8-s + 0.469i·9-s + (−0.261 − 1.63i)10-s + (−0.149 − 0.149i)11-s + (0.650 + 0.327i)12-s + (0.300 + 0.300i)13-s + (−0.713 + 0.985i)14-s − 1.20i·15-s + (−0.803 + 0.594i)16-s − 0.774·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.515i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.857 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.167025 + 0.602095i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.167025 + 0.602095i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.34 - 3.24i)T \) |
good | 3 | \( 1 + (4.63 - 4.63i)T - 81iT^{2} \) |
| 5 | \( 1 + (29.2 - 29.2i)T - 625iT^{2} \) |
| 7 | \( 1 - 59.6T + 2.40e3T^{2} \) |
| 11 | \( 1 + (18.0 + 18.0i)T + 1.46e4iT^{2} \) |
| 13 | \( 1 + (-50.7 - 50.7i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + 223.T + 8.35e4T^{2} \) |
| 19 | \( 1 + (-14.7 + 14.7i)T - 1.30e5iT^{2} \) |
| 23 | \( 1 - 739.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-938. - 938. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 - 938. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + (-263. + 263. i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 + 248. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (-1.03e3 - 1.03e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 2.01e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (-833. + 833. i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + (2.22e3 + 2.22e3i)T + 1.21e7iT^{2} \) |
| 61 | \( 1 + (341. + 341. i)T + 1.38e7iT^{2} \) |
| 67 | \( 1 + (-4.84e3 + 4.84e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 + 4.18e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 9.07e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 735. iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-1.44e3 + 1.44e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 - 5.07e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 2.52e3T + 8.85e7T^{2} \) |
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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
−18.64234965369657600187396550767, −17.59314611754506436783854474420, −16.15229513641928954249784619848, −15.22052114135907929733526234869, −14.17082601783221074052722958633, −11.22564807435143651269661579608, −10.72986400653809419155181796425, −8.368308337485112717115875153783, −7.02113534111111739207514274888, −4.79570633276403909130541501185,
0.865822807810832524338782692320, 4.50242342143525858189527350076, 7.70603471298005074132102157777, 8.851294390512199938649562402821, 11.19421814224389576821305724521, 11.97597112646444880793895846938, 13.07426441537356061653501906523, 15.43997972945895984241752992716, 17.00205245477774661125487578705, 17.84424831119583119878803806045