| L(s) = 1 | + 11i·7-s − 23i·13-s + 37·19-s − 46·31-s − 73i·37-s + 22i·43-s − 72·49-s + 47·61-s − 13i·67-s − 143i·73-s − 11·79-s + 253·91-s − 169i·97-s + 157i·103-s + 214·109-s + ⋯ |
| L(s) = 1 | + 1.57i·7-s − 1.76i·13-s + 1.94·19-s − 1.48·31-s − 1.97i·37-s + 0.511i·43-s − 1.46·49-s + 0.770·61-s − 0.194i·67-s − 1.95i·73-s − 0.139·79-s + 2.78·91-s − 1.74i·97-s + 1.52i·103-s + 1.96·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.947015626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.947015626\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 11iT - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 23iT - 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 - 37T + 361T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 46T + 961T^{2} \) |
| 37 | \( 1 + 73iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 22iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 47T + 3.72e3T^{2} \) |
| 67 | \( 1 + 13iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 143iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 11T + 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 169iT - 9.40e3T^{2} \) |
| 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
−8.693029164915622428680294124128, −7.79418755598108967235016244840, −7.31492429688172838923953167701, −5.96063484352738532636700755573, −5.57370813926779999821188003884, −5.00438938029269444775855464466, −3.51695428250784421167031966582, −2.92506605067315971924054493644, −1.93872028989534773688614048140, −0.55998066346963596578799167088,
0.913405412043670330867599824668, 1.76915795043288088701492266856, 3.20782584247536254999097402133, 3.95636778317204524316573817965, 4.66288401847522957918206969994, 5.56486949655109923383470123456, 6.79106316613139715418890348559, 7.05555223400700319680050632608, 7.80089769820038421778627656763, 8.758968719040413724159401856961
