L(s) = 1 | − 18.5·7-s + 29.3i·11-s − 181. i·23-s − 125·25-s + 304. i·29-s − 10.5·37-s + 534.·43-s + 343.·49-s − 768. i·53-s − 740·67-s − 1.17e3i·71-s − 544. i·77-s + 1.38e3·79-s + 20.9i·107-s + 2.27e3·109-s + ⋯ |
L(s) = 1 | − 0.999·7-s + 0.805i·11-s − 1.64i·23-s − 25-s + 1.94i·29-s − 0.0470·37-s + 1.89·43-s + 1.00·49-s − 1.99i·53-s − 1.34·67-s − 1.97i·71-s − 0.805i·77-s + 1.97·79-s + 0.0189i·107-s + 1.99·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.261932947\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.261932947\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 18.5T \) |
good | 5 | \( 1 + 125T^{2} \) |
| 11 | \( 1 - 29.3iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.85e3T^{2} \) |
| 23 | \( 1 + 181. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 304. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 2.97e4T^{2} \) |
| 37 | \( 1 + 10.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 534.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 + 768. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 + 740T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.17e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.38e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 - 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.442363247041037842508732239702, −8.789947084914737766536826013706, −7.69507563290555552287076237445, −6.86299330010372017822516675013, −6.18211933768406519016636796171, −5.08069720115568175951651311484, −4.09449751545064842404715346439, −3.07218059553523711674112592568, −1.97691888578098271903961341539, −0.41056246105456451057362411569,
0.838009708393414333374423632121, 2.38559447445924050929783578172, 3.43230304529342847605840724172, 4.23938782130990709361146995833, 5.77386582626608236300011184105, 6.02817261024154411107435213069, 7.28534157261273765556800348520, 7.943873676624762202920086627845, 9.088064204253452117370582049935, 9.603212428133466514063835845623