L(s) = 1 | + i·2-s − 4-s + (1.75 + 0.172i)5-s − i·8-s + (0.195 + 0.980i)9-s + (−0.172 + 1.75i)10-s + (−1.63 − 1.08i)13-s + 16-s + (1.17 + 1.17i)17-s + (−0.980 + 0.195i)18-s + (−1.75 − 0.172i)20-s + (2.07 + 0.411i)25-s + (1.08 − 1.63i)26-s + (0.275 − 1.38i)29-s + i·32-s + ⋯ |
L(s) = 1 | + i·2-s − 4-s + (1.75 + 0.172i)5-s − i·8-s + (0.195 + 0.980i)9-s + (−0.172 + 1.75i)10-s + (−1.63 − 1.08i)13-s + 16-s + (1.17 + 1.17i)17-s + (−0.980 + 0.195i)18-s + (−1.75 − 0.172i)20-s + (2.07 + 0.411i)25-s + (1.08 − 1.63i)26-s + (0.275 − 1.38i)29-s + i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0543 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0543 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.195983737\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.195983737\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 257 | \( 1 + T \) |
good | 3 | \( 1 + (-0.195 - 0.980i)T^{2} \) |
| 5 | \( 1 + (-1.75 - 0.172i)T + (0.980 + 0.195i)T^{2} \) |
| 7 | \( 1 + (0.831 + 0.555i)T^{2} \) |
| 11 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (1.63 + 1.08i)T + (0.382 + 0.923i)T^{2} \) |
| 17 | \( 1 + (-1.17 - 1.17i)T + iT^{2} \) |
| 19 | \( 1 + (0.555 - 0.831i)T^{2} \) |
| 23 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.275 + 1.38i)T + (-0.923 - 0.382i)T^{2} \) |
| 31 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 37 | \( 1 + (1.68 - 0.512i)T + (0.831 - 0.555i)T^{2} \) |
| 41 | \( 1 + (0.172 + 0.0924i)T + (0.555 + 0.831i)T^{2} \) |
| 43 | \( 1 + (0.195 - 0.980i)T^{2} \) |
| 47 | \( 1 + (0.555 - 0.831i)T^{2} \) |
| 53 | \( 1 + (-0.0924 - 0.938i)T + (-0.980 + 0.195i)T^{2} \) |
| 59 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 61 | \( 1 + (0.636 + 0.425i)T + (0.382 + 0.923i)T^{2} \) |
| 67 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 71 | \( 1 + (0.555 + 0.831i)T^{2} \) |
| 73 | \( 1 + (0.149 - 0.360i)T + (-0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 83 | \( 1 + (0.195 - 0.980i)T^{2} \) |
| 89 | \( 1 + (-0.216 - 0.324i)T + (-0.382 + 0.923i)T^{2} \) |
| 97 | \( 1 + (-0.124 + 1.26i)T + (-0.980 - 0.195i)T^{2} \) |
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
−9.979310699679220345022125565816, −9.801503453350175679852620314410, −8.512704714484017527292469044138, −7.78900125502211549583741121805, −6.96392982068639468579323552854, −5.90348583613563610142928354928, −5.44546474334347280209394132388, −4.67852473723154581115201394965, −3.02355117026895088053723520843, −1.78372075867876704961122071100,
1.37828472835755954226645719106, 2.37532307034652720143064142655, 3.36486195655928157787210274108, 4.86131278208250810592220373818, 5.29356996044291923697629528919, 6.47861535646834333911486143956, 7.35567292295286412448060969693, 8.930672575222785415559444263279, 9.320740764474637625981604597843, 9.903073542732809715588608717237