| L(s) = 1 | + (−1.90 + 4.59i)3-s + (−0.188 + 0.0782i)5-s + (11.4 − 11.4i)7-s + (1.63 + 1.63i)9-s + (18.6 + 45.0i)11-s + (−18.9 − 7.84i)13-s − 1.01i·15-s − 85.7i·17-s + (110. + 45.8i)19-s + (30.6 + 74.0i)21-s + (74.2 + 74.2i)23-s + (−88.3 + 88.3i)25-s + (−134. + 55.7i)27-s + (−64.4 + 155. i)29-s − 36.6·31-s + ⋯ |
| L(s) = 1 | + (−0.365 + 0.883i)3-s + (−0.0168 + 0.00699i)5-s + (0.616 − 0.616i)7-s + (0.0603 + 0.0603i)9-s + (0.511 + 1.23i)11-s + (−0.404 − 0.167i)13-s − 0.0174i·15-s − 1.22i·17-s + (1.33 + 0.553i)19-s + (0.318 + 0.769i)21-s + (0.672 + 0.672i)23-s + (−0.706 + 0.706i)25-s + (−0.958 + 0.397i)27-s + (−0.412 + 0.996i)29-s − 0.212·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.588784507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.588784507\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (1.90 - 4.59i)T + (-19.0 - 19.0i)T^{2} \) |
| 5 | \( 1 + (0.188 - 0.0782i)T + (88.3 - 88.3i)T^{2} \) |
| 7 | \( 1 + (-11.4 + 11.4i)T - 343iT^{2} \) |
| 11 | \( 1 + (-18.6 - 45.0i)T + (-941. + 941. i)T^{2} \) |
| 13 | \( 1 + (18.9 + 7.84i)T + (1.55e3 + 1.55e3i)T^{2} \) |
| 17 | \( 1 + 85.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (-110. - 45.8i)T + (4.85e3 + 4.85e3i)T^{2} \) |
| 23 | \( 1 + (-74.2 - 74.2i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + (64.4 - 155. i)T + (-1.72e4 - 1.72e4i)T^{2} \) |
| 31 | \( 1 + 36.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + (313. - 129. i)T + (3.58e4 - 3.58e4i)T^{2} \) |
| 41 | \( 1 + (-196. - 196. i)T + 6.89e4iT^{2} \) |
| 43 | \( 1 + (20.8 + 50.4i)T + (-5.62e4 + 5.62e4i)T^{2} \) |
| 47 | \( 1 - 508. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (73.8 + 178. i)T + (-1.05e5 + 1.05e5i)T^{2} \) |
| 59 | \( 1 + (-40.9 + 16.9i)T + (1.45e5 - 1.45e5i)T^{2} \) |
| 61 | \( 1 + (-324. + 784. i)T + (-1.60e5 - 1.60e5i)T^{2} \) |
| 67 | \( 1 + (49.4 - 119. i)T + (-2.12e5 - 2.12e5i)T^{2} \) |
| 71 | \( 1 + (-362. + 362. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (-239. - 239. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 - 1.01e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-231. - 95.7i)T + (4.04e5 + 4.04e5i)T^{2} \) |
| 89 | \( 1 + (-1.10e3 + 1.10e3i)T - 7.04e5iT^{2} \) |
| 97 | \( 1 + 74.0T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.59292044465929692947079867159, −10.93307795112072381304205560687, −9.748300333678502230985221923644, −9.465502290784181384923609996758, −7.65537019110399737348132625818, −7.12769634450123805054407529930, −5.28027205886666459799780007043, −4.72897365683879521246930623602, −3.49633757262252579963467213503, −1.50762029029422897670460072494,
0.71289226682364925028417110686, 2.09075414737137470726037507271, 3.78777710543343367854157820894, 5.39587478058117182475062391423, 6.23549657264626931759216974296, 7.28859962416336582572196732695, 8.330789448683984546368820336390, 9.195389134249687084203776360891, 10.54624737316403706058215425448, 11.67272557289200142700180693719
