L(s) = 1 | + (−3.96 − 2.28i)2-s + (−3.96 + 3.36i)3-s + (6.45 + 11.1i)4-s + (23.3 − 4.26i)6-s + (−17.4 − 10.0i)7-s − 22.4i·8-s + (4.37 − 26.6i)9-s + (−33.1 + 57.4i)11-s + (−63.2 − 22.5i)12-s + (−40.5 + 23.4i)13-s + (45.9 + 79.6i)14-s + (0.237 − 0.411i)16-s + 47.6i·17-s + (−78.2 + 95.5i)18-s + 9.95·19-s + ⋯ |
L(s) = 1 | + (−1.40 − 0.808i)2-s + (−0.762 + 0.647i)3-s + (0.807 + 1.39i)4-s + (1.59 − 0.290i)6-s + (−0.940 − 0.543i)7-s − 0.993i·8-s + (0.162 − 0.986i)9-s + (−0.909 + 1.57i)11-s + (−1.52 − 0.543i)12-s + (−0.864 + 0.499i)13-s + (0.878 + 1.52i)14-s + (0.00371 − 0.00643i)16-s + 0.679i·17-s + (−1.02 + 1.25i)18-s + 0.120·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.162177 - 0.142886i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.162177 - 0.142886i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (3.96 - 3.36i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (3.96 + 2.28i)T + (4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (17.4 + 10.0i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (33.1 - 57.4i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (40.5 - 23.4i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 47.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 9.95T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-8.30 + 4.79i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (89.3 - 154. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (77.0 + 133. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 248. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + (124. + 216. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-183. - 106. i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-411. - 237. i)T + (5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 546. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (209. + 363. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-272. + 472. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-387. + 223. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 409.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 358. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (325. - 564. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-704. - 406. i)T + (2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 200.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-218. - 126. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.13963221416478012480958422071, −10.42758868830739448887944420333, −9.748576934122751381069200350890, −9.243587489121576350665152274067, −7.62892558380576953856525119456, −6.83994067985770868660569574804, −5.16554351292150656769542992383, −3.76491856448478235070325835584, −2.14226380213553347537707675049, −0.25724540645996849083565586677,
0.69054609298924393223449848816, 2.74962982987454283200524001505, 5.43339106501066683629015570736, 6.07179508261047286516435117047, 7.14068540844589830965142157380, 7.948483354543834594902742084311, 8.916394935969489158371538877264, 9.991213706103682044499490843676, 10.76398250379792758028339592083, 11.85017170856885657632942475318