| L(s) = 1 | + 2.82i·2-s − 8.00·4-s − 11.1i·5-s + 47.7·7-s − 22.6i·8-s + 31.6·10-s − 194. i·11-s + 272.·13-s + 134. i·14-s + 64.0·16-s − 363. i·17-s − 369.·19-s + 89.4i·20-s + 551.·22-s − 745. i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 0.447i·5-s + 0.973·7-s − 0.353i·8-s + 0.316·10-s − 1.61i·11-s + 1.61·13-s + 0.688i·14-s + 0.250·16-s − 1.25i·17-s − 1.02·19-s + 0.223i·20-s + 1.13·22-s − 1.40i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.693445627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.693445627\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 11.1iT \) |
| good | 7 | \( 1 - 47.7T + 2.40e3T^{2} \) |
| 11 | \( 1 + 194. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 272.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 363. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 369.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 745. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 35.1iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 503.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.32e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.93e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 988.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.08e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.73e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 3.93e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 939.T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.76e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 3.22e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 8.55e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 2.19e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 5.40e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 7.38e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 9.64e3T + 8.85e7T^{2} \) |
| 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
−8.987551693787820835101175970126, −8.492317730062601523994315972684, −8.069021734037138732358509456889, −6.70220286971787331559677644721, −6.01133581192898804700822100283, −5.07707863842217430772353972994, −4.23378806752313961662619860789, −3.05794175687724242445877168870, −1.38012392513184694005920404574, −0.38933353212001111932179916755,
1.52964044796429832594050111487, 1.95962342821031196233057076958, 3.55784122124815516910998487735, 4.24844010392142384667787655361, 5.31561892577300638439932333339, 6.38520518908234022619673718211, 7.43492950328654419212033672085, 8.344852269369086129356183113112, 9.015875285218129562380664165542, 10.26412067753799143943618422101
