| L(s) = 1 | − 27i·3-s + (231. − 156. i)5-s + 536. i·7-s − 729·9-s − 5.83e3·11-s + 5.81e3i·13-s + (−4.22e3 − 6.25e3i)15-s + 9.47e3i·17-s + 5.29e4·19-s + 1.44e4·21-s − 7.49e4i·23-s + (2.91e4 − 7.24e4i)25-s + 1.96e4i·27-s + 2.21e4·29-s + 1.58e5·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.828 − 0.559i)5-s + 0.590i·7-s − 0.333·9-s − 1.32·11-s + 0.734i·13-s + (−0.323 − 0.478i)15-s + 0.467i·17-s + 1.77·19-s + 0.341·21-s − 1.28i·23-s + (0.373 − 0.927i)25-s + 0.192i·27-s + 0.168·29-s + 0.957·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.327639503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.327639503\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 27iT \) |
| 5 | \( 1 + (-231. + 156. i)T \) |
| good | 7 | \( 1 - 536. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 5.83e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.81e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 9.47e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 5.29e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.49e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 2.21e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.58e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.64e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 7.71e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.45e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 8.93e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 1.53e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.55e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 7.21e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.87e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 1.89e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.67e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 2.70e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.54e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 1.17e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.83e6iT - 8.07e13T^{2} \) |
| 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
−10.65295479411813727638876106062, −9.701837542802076626743990011009, −8.728607801173308999949407537931, −7.87562232312959945880826220185, −6.60484403172589481229260682695, −5.62169358163787187224141856542, −4.78834883515710360133738464642, −2.87269269985851369308011334808, −1.93964778752274702059954642987, −0.68517457059393707825836428736,
0.911157131013633821306333179760, 2.58539336207023373242128794283, 3.43701996884037934270915764688, 5.07677020819486798668718811664, 5.68624867459765190004607984178, 7.15004189183999456255338580451, 7.956343929548539177652295893340, 9.461903180415349300998656149275, 10.06501737212915521179572088803, 10.78100668139182522134720268966
