| L(s) = 1 | + (3.55 + 1.15i)3-s + (10.9 − 2.10i)5-s + 30.1i·7-s + (−10.5 − 7.66i)9-s + (51.5 − 37.4i)11-s + (−23.8 + 32.8i)13-s + (41.4 + 5.21i)15-s + (42.1 − 13.6i)17-s + (35.4 + 109. i)19-s + (−34.7 + 107. i)21-s + (−14.3 − 19.7i)23-s + (116. − 46.1i)25-s + (−87.9 − 121. i)27-s + (2.06 − 6.36i)29-s + (−97.5 − 300. i)31-s + ⋯ |
| L(s) = 1 | + (0.683 + 0.222i)3-s + (0.982 − 0.187i)5-s + 1.62i·7-s + (−0.390 − 0.283i)9-s + (1.41 − 1.02i)11-s + (−0.509 + 0.700i)13-s + (0.713 + 0.0897i)15-s + (0.601 − 0.195i)17-s + (0.427 + 1.31i)19-s + (−0.361 + 1.11i)21-s + (−0.130 − 0.179i)23-s + (0.929 − 0.369i)25-s + (−0.626 − 0.862i)27-s + (0.0132 − 0.0407i)29-s + (−0.565 − 1.74i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.482i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.875 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.13926 + 0.550501i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.13926 + 0.550501i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-10.9 + 2.10i)T \) |
| good | 3 | \( 1 + (-3.55 - 1.15i)T + (21.8 + 15.8i)T^{2} \) |
| 7 | \( 1 - 30.1iT - 343T^{2} \) |
| 11 | \( 1 + (-51.5 + 37.4i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (23.8 - 32.8i)T + (-678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-42.1 + 13.6i)T + (3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-35.4 - 109. i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (14.3 + 19.7i)T + (-3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-2.06 + 6.36i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (97.5 + 300. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (201. - 277. i)T + (-1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (356. + 259. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 291. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-100. - 32.7i)T + (8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (346. + 112. i)T + (1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (269. + 195. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-10.8 + 7.90i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-54.7 + 17.7i)T + (2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (72.3 - 222. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-262. - 360. i)T + (-1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (17.4 - 53.8i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-1.29e3 + 421. i)T + (4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (426. - 309. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (1.27e3 + 413. i)T + (7.38e5 + 5.36e5i)T^{2} \) |
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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
−13.85741008976797911009910985775, −12.22769553329071193900736940162, −11.67771054441583850648927591629, −9.778988742186268414145392888596, −9.119201276689612186221168194915, −8.405256287843285957105274613883, −6.30960608388160671495266886229, −5.52023874121411238708818353454, −3.44350660851471538219888826075, −1.98523379101894069062422548760,
1.50643644213445597494405324444, 3.28053967058540001022613797614, 4.94599862948718857037730467266, 6.75682663926353111886932562336, 7.52093376656711271451872388304, 9.076415832025153249057685146813, 9.992186218375082968605420367088, 10.95550577920958395880949100729, 12.51595497083045118431602405551, 13.60163782531980062129719277167
