| L(s) = 1 | + (11.3 − 0.338i)2-s + (127. − 7.64i)4-s − 93.0·5-s + 1.02e3i·7-s + (1.44e3 − 129. i)8-s + (−1.05e3 + 31.4i)10-s + 1.54e3i·11-s + 5.67e3i·13-s + (346. + 1.15e4i)14-s + (1.62e4 − 1.95e3i)16-s + 1.57e4i·17-s + 3.39e4·19-s + (−1.18e4 + 712. i)20-s + (522. + 1.74e4i)22-s + 3.00e4·23-s + ⋯ |
| L(s) = 1 | + (0.999 − 0.0298i)2-s + (0.998 − 0.0597i)4-s − 0.333·5-s + 1.12i·7-s + (0.995 − 0.0895i)8-s + (−0.332 + 0.00995i)10-s + 0.350i·11-s + 0.716i·13-s + (0.0337 + 1.12i)14-s + (0.992 − 0.119i)16-s + 0.775i·17-s + 1.13·19-s + (−0.332 + 0.0199i)20-s + (0.0104 + 0.349i)22-s + 0.515·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.864i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.501 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.00033 + 1.72788i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.00033 + 1.72788i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-11.3 + 0.338i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 93.0T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.02e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 1.54e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 5.67e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 1.57e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 3.39e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.00e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.90e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.52e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 3.36e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 6.63e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + 7.07e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.02e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.62e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.74e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 3.21e5iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 4.64e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.53e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.41e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.60e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + 8.11e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 1.00e7iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 5.88e6T + 8.07e13T^{2} \) |
| 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
−13.36544102952728386411911109959, −12.12232312908753216088276106064, −11.69417084755167128020263515887, −10.21090829309622627024898418664, −8.693925956111333384992025237382, −7.23744625738074521118106903338, −5.95589912171926263972731017990, −4.76459477617868618611922679517, −3.27003179111992578158645653871, −1.80443028295640526353800033751,
0.905203164155773042494369026712, 2.99608838969033424008409391443, 4.19208984692467437423351313986, 5.52846874777670372199325050047, 7.03262979293137394538988226905, 7.904221940495056451449351374531, 9.921085224513151394578138681113, 11.05025571507230969271063927512, 11.94242103981684312714473967915, 13.29098599305722270746267524767
