| L(s) = 1 | + (−1 − 2.82i)3-s + 5.65i·5-s + (−7.00 + 5.65i)9-s − 5.65i·11-s − 10·13-s + (16.0 − 5.65i)15-s − 22.6i·17-s − 2·19-s + 11.3i·23-s − 7.00·25-s + (23.0 + 14.1i)27-s − 16.9i·29-s + 22·31-s + (−16.0 + 5.65i)33-s − 6·37-s + ⋯ |
| L(s) = 1 | + (−0.333 − 0.942i)3-s + 1.13i·5-s + (−0.777 + 0.628i)9-s − 0.514i·11-s − 0.769·13-s + (1.06 − 0.377i)15-s − 1.33i·17-s − 0.105·19-s + 0.491i·23-s − 0.280·25-s + (0.851 + 0.523i)27-s − 0.585i·29-s + 0.709·31-s + (−0.484 + 0.171i)33-s − 0.162·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.349569912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.349569912\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 + 2.82i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 5.65iT - 25T^{2} \) |
| 11 | \( 1 + 5.65iT - 121T^{2} \) |
| 13 | \( 1 + 10T + 169T^{2} \) |
| 17 | \( 1 + 22.6iT - 289T^{2} \) |
| 19 | \( 1 + 2T + 361T^{2} \) |
| 23 | \( 1 - 11.3iT - 529T^{2} \) |
| 29 | \( 1 + 16.9iT - 841T^{2} \) |
| 31 | \( 1 - 22T + 961T^{2} \) |
| 37 | \( 1 + 6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 33.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 82T + 1.84e3T^{2} \) |
| 47 | \( 1 - 67.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 62.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 73.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 86T + 3.72e3T^{2} \) |
| 67 | \( 1 - 2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 124. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 82T + 5.32e3T^{2} \) |
| 79 | \( 1 - 10T + 6.24e3T^{2} \) |
| 83 | \( 1 + 73.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 33.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 94T + 9.40e3T^{2} \) |
| 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
−9.705742369557791484116920636740, −8.723734930495159636561748606810, −7.55221161419253050400809773489, −7.30453261614245276068656867358, −6.34641495465019141315356431912, −5.66356060041287039042793732552, −4.52325161320778741248828865678, −2.96863840142697748307446282849, −2.48269222113906623690332212663, −0.874890692367337895974216340922,
0.56130893396532418577808620310, 2.10847724879635944579758693494, 3.60454640769073190694486672655, 4.48790621766800915413295055725, 5.08426713406861487908877024997, 5.93464170940541968101130444680, 6.98244319821467629846908490729, 8.223967026506504283713956496418, 8.772914202252505087665193009923, 9.575697881950485442702339700806
