| L(s) = 1 | + (−2.49 − 1.33i)2-s + (4.10 + 0.724i)3-s + (4.44 + 6.65i)4-s + (9.57 − 11.4i)5-s + (−9.28 − 7.28i)6-s + (−5.68 − 9.85i)7-s + (−2.20 − 22.5i)8-s + (−9.01 − 3.27i)9-s + (−39.0 + 15.6i)10-s + (−7.59 − 4.38i)11-s + (13.4 + 30.5i)12-s + (28.0 − 4.94i)13-s + (1.04 + 32.1i)14-s + (47.6 − 39.9i)15-s + (−24.5 + 59.1i)16-s + (61.2 − 22.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.881 − 0.471i)2-s + (0.790 + 0.139i)3-s + (0.555 + 0.831i)4-s + (0.856 − 1.02i)5-s + (−0.631 − 0.495i)6-s + (−0.307 − 0.531i)7-s + (−0.0972 − 0.995i)8-s + (−0.333 − 0.121i)9-s + (−1.23 + 0.495i)10-s + (−0.208 − 0.120i)11-s + (0.323 + 0.735i)12-s + (0.597 − 0.105i)13-s + (0.0199 + 0.613i)14-s + (0.819 − 0.687i)15-s + (−0.383 + 0.923i)16-s + (0.873 − 0.317i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.181 + 0.983i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.927202 - 1.11434i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.927202 - 1.11434i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.49 + 1.33i)T \) |
| 19 | \( 1 + (72.9 + 39.2i)T \) |
| good | 3 | \( 1 + (-4.10 - 0.724i)T + (25.3 + 9.23i)T^{2} \) |
| 5 | \( 1 + (-9.57 + 11.4i)T + (-21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (5.68 + 9.85i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (7.59 + 4.38i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-28.0 + 4.94i)T + (2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (-61.2 + 22.2i)T + (3.76e3 - 3.15e3i)T^{2} \) |
| 23 | \( 1 + (-57.3 + 48.1i)T + (2.11e3 - 1.19e4i)T^{2} \) |
| 29 | \( 1 + (34.5 - 95.0i)T + (-1.86e4 - 1.56e4i)T^{2} \) |
| 31 | \( 1 + (26.3 + 45.6i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 113. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + (-52.0 + 295. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-115. + 138. i)T + (-1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-147. - 53.7i)T + (7.95e4 + 6.67e4i)T^{2} \) |
| 53 | \( 1 + (-356. - 425. i)T + (-2.58e4 + 1.46e5i)T^{2} \) |
| 59 | \( 1 + (-72.6 - 199. i)T + (-1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-273. - 326. i)T + (-3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (216. - 594. i)T + (-2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-223. - 187. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 + (-110. + 626. i)T + (-3.65e5 - 1.33e5i)T^{2} \) |
| 79 | \( 1 + (138. - 785. i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (843. - 486. i)T + (2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-214. - 1.21e3i)T + (-6.62e5 + 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-265. + 96.7i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−12.33702458257948324291769863180, −10.94570675214062041220135454796, −9.992653096817125263123511184387, −8.987543336507057812609678356947, −8.602202138455579692305307704017, −7.25480893164620932108322547781, −5.71846820197488878818420875894, −3.86149055899742904644178793396, −2.46340323352084070652764865741, −0.852307875988986622420546201721,
1.95099723549596736166903422555, 3.04260272821413287501148204644, 5.67621415681693321324062627181, 6.44235822271497417405863191456, 7.71509661479826521996240891557, 8.654187970439435168304862646788, 9.626924122907669958305171045516, 10.45487024007663781249448869310, 11.45829711421906639419157146710, 13.06582018127140135026995226968
