L(s) = 1 | − 2.82i·2-s − 15.3i·3-s − 8.00·4-s + 41.6·5-s − 43.4·6-s − 62.4·7-s + 22.6i·8-s − 154.·9-s − 117. i·10-s + 122.·11-s + 122. i·12-s + 68.9i·13-s + 176. i·14-s − 638. i·15-s + 64.0·16-s + 297.·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 1.70i·3-s − 0.500·4-s + 1.66·5-s − 1.20·6-s − 1.27·7-s + 0.353i·8-s − 1.90·9-s − 1.17i·10-s + 1.01·11-s + 0.852i·12-s + 0.408i·13-s + 0.900i·14-s − 2.83i·15-s + 0.250·16-s + 1.02·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.436367 - 1.48188i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.436367 - 1.48188i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2.82iT \) |
| 19 | \( 1 + (195. + 303. i)T \) |
good | 3 | \( 1 + 15.3iT - 81T^{2} \) |
| 5 | \( 1 - 41.6T + 625T^{2} \) |
| 7 | \( 1 + 62.4T + 2.40e3T^{2} \) |
| 11 | \( 1 - 122.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 68.9iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 297.T + 8.35e4T^{2} \) |
| 23 | \( 1 - 268.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 561. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 252. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 2.40e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 690. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 218.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 83.1T + 4.87e6T^{2} \) |
| 53 | \( 1 - 4.38e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 476. iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 3.96e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 5.11e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 8.18e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 7.34e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.48e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 5.34e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 1.11e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.07e3iT - 8.85e7T^{2} \) |
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
−14.34570111565171575480150988690, −13.44015980082312426086863214634, −12.90124739540166186329210879024, −11.78727045842362205483605195228, −9.964953947314055989434160460092, −8.940874543636569378318745302429, −6.82087467934725025219984456751, −5.97416401507610809860735011924, −2.66338611845239073724008034363, −1.22290137124919347744828337817,
3.51547206206605362420800676381, 5.38000098369012329067096369423, 6.33767453472833263068159137589, 9.027302035996747475320046176994, 9.697564380039864465319740877324, 10.44619371237989371549386591668, 12.74665091902373497643016870598, 14.14984035605564647110187196846, 14.84719777626063257991396169105, 16.34057875873229856748284081765