L(s) = 1 | + (−1.51 − 2.38i)2-s + (−3.38 + 7.24i)4-s − 13.1i·5-s − 4.49i·7-s + (22.4 − 2.92i)8-s + (−31.4 + 20.0i)10-s − 22.3·11-s − 73.5·13-s + (−10.7 + 6.81i)14-s + (−41.0 − 49.1i)16-s − 42.6i·17-s + 122. i·19-s + (95.6 + 44.7i)20-s + (33.9 + 53.3i)22-s − 197.·23-s + ⋯ |
L(s) = 1 | + (−0.536 − 0.843i)2-s + (−0.423 + 0.905i)4-s − 1.18i·5-s − 0.242i·7-s + (0.991 − 0.129i)8-s + (−0.995 + 0.633i)10-s − 0.612·11-s − 1.56·13-s + (−0.204 + 0.130i)14-s + (−0.641 − 0.767i)16-s − 0.608i·17-s + 1.47i·19-s + (1.06 + 0.499i)20-s + (0.328 + 0.516i)22-s − 1.79·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.100807 + 0.453699i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.100807 + 0.453699i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.51 + 2.38i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 13.1iT - 125T^{2} \) |
| 7 | \( 1 + 4.49iT - 343T^{2} \) |
| 11 | \( 1 + 22.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 73.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 42.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 122. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 197.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 14.6iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 147. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 234.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 396. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 280. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 534.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 337. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 672.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 80.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 251. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 95.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 251.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 499. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 16.1T + 5.71e5T^{2} \) |
| 89 | \( 1 - 321. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 210.T + 9.12e5T^{2} \) |
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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
−12.37790298742563417471202283980, −11.82274394518426878739969787705, −10.23233163521911411090228500550, −9.625660040261528309688393723729, −8.339019613887686014935143447862, −7.52040831500939099157938011744, −5.29795142685982665320218437640, −4.07821942464880267484139673298, −2.13797049366818692189561971823, −0.28910281676697173826437296512,
2.53346962644923761559213104593, 4.76481364762127025197203485397, 6.19330570728805026350496955696, 7.18923600271869084016677010456, 8.128642560735445890513370894493, 9.623700699131690893608242136043, 10.35764361868191430653974694796, 11.46346673348884005223846145286, 12.99161268455469738755032259379, 14.24287561661442316167694017994