L(s) = 1 | + 24.1i·3-s + (−46.7 + 30.6i)5-s − 179. i·7-s − 339.·9-s − 653.·11-s − 284. i·13-s + (−740. − 1.12e3i)15-s + 383. i·17-s + 2.56e3·19-s + 4.34e3·21-s + 948. i·23-s + (1.24e3 − 2.86e3i)25-s − 2.33e3i·27-s − 1.52e3·29-s − 3.10e3·31-s + ⋯ |
L(s) = 1 | + 1.54i·3-s + (−0.836 + 0.548i)5-s − 1.38i·7-s − 1.39·9-s − 1.62·11-s − 0.467i·13-s + (−0.849 − 1.29i)15-s + 0.321i·17-s + 1.62·19-s + 2.14·21-s + 0.373i·23-s + (0.398 − 0.917i)25-s − 0.615i·27-s − 0.336·29-s − 0.580·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.161262448\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.161262448\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (46.7 - 30.6i)T \) |
good | 3 | \( 1 - 24.1iT - 243T^{2} \) |
| 7 | \( 1 + 179. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 653.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 284. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 383. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.56e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 948. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 1.52e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.10e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.99e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.51e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.75e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.47e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 8.70e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.26e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.30e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.61e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.62e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.13e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.93e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.95e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.30e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.62e4iT - 8.58e9T^{2} \) |
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
−10.72694261995924987206647105530, −10.23619659110538056178906680714, −9.290865834239280826977931253036, −7.76817410595285541313024298717, −7.46414682844236597571546950154, −5.64283446506670864891004842942, −4.65213470164039447468350705394, −3.73523338293967947581854227992, −2.99006423371546851500093409344, −0.47902289191581290682935811742,
0.70937336679334322433829747808, 2.07884099517618706255970374519, 3.04641866355727704039653927270, 5.01866794829273094048530039371, 5.75891997683435960771017377518, 7.09567307872275444431064484220, 7.83825862130286452789691270869, 8.487854338137981655923209997055, 9.517995176521661912332012881629, 11.15876952573403462470509368269