L(s) = 1 | + 4.88i·2-s − 3i·3-s − 15.9·4-s + (−9.63 − 5.67i)5-s + 14.6·6-s − 7i·7-s − 38.6i·8-s − 9·9-s + (27.7 − 47.0i)10-s + 54.9·11-s + 47.7i·12-s − 49.7i·13-s + 34.2·14-s + (−17.0 + 28.8i)15-s + 61.7·16-s − 133. i·17-s + ⋯ |
L(s) = 1 | + 1.72i·2-s − 0.577i·3-s − 1.98·4-s + (−0.861 − 0.508i)5-s + 0.998·6-s − 0.377i·7-s − 1.70i·8-s − 0.333·9-s + (0.878 − 1.48i)10-s + 1.50·11-s + 1.14i·12-s − 1.06i·13-s + 0.653·14-s + (−0.293 + 0.497i)15-s + 0.964·16-s − 1.90i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.508i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.861 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.755842 - 0.206288i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.755842 - 0.206288i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3iT \) |
| 5 | \( 1 + (9.63 + 5.67i)T \) |
| 7 | \( 1 + 7iT \) |
good | 2 | \( 1 - 4.88iT - 8T^{2} \) |
| 11 | \( 1 - 54.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 133. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 138.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 7.32iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 87.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 209.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 67.9iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 77.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 197. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 4.97iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 53.0iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 683.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 26.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 149. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 6.15T + 3.57e5T^{2} \) |
| 73 | \( 1 - 294. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 938.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 784. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 275.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.16e3iT - 9.12e5T^{2} \) |
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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
−13.45599382062213503484712799753, −12.48698607219363729038366796433, −11.27743264105076338988419045449, −9.304928023744878854175232108392, −8.442808553472678563399484452737, −7.42330824102590640755743711813, −6.69644002191792149335098483460, −5.30095993833632765420276781059, −3.99974671043869865632450468572, −0.44928666203797719394675372730,
1.93752015487799232672000412063, 3.77129916583061970201418224630, 4.19181882194962807505477648783, 6.44236979367788911178769130853, 8.524443798674431410728837745220, 9.275927268385660273364778970614, 10.57109110606400198242669912792, 11.20932378825912725843916749142, 12.03772515401801868329612376679, 12.88700374079946699487414139962