| L(s) = 1 | + (−5.96 − 4.75i)2-s + (−1.39 − 9.26i)3-s + (9.39 + 41.1i)4-s + (22.1 + 32.5i)5-s + (−35.7 + 61.8i)6-s + (−17.4 + 10.0i)7-s + (86.7 − 180. i)8-s + (−6.46 + 1.99i)9-s + (22.4 − 299. i)10-s + (−43.1 + 189. i)11-s + (368. − 144. i)12-s + (2.56 + 34.1i)13-s + (152. + 22.9i)14-s + (270. − 251. i)15-s + (−765. + 368. i)16-s + (115. + 78.7i)17-s + ⋯ |
| L(s) = 1 | + (−1.49 − 1.18i)2-s + (−0.155 − 1.02i)3-s + (0.586 + 2.57i)4-s + (0.887 + 1.30i)5-s + (−0.992 + 1.71i)6-s + (−0.356 + 0.205i)7-s + (1.35 − 2.81i)8-s + (−0.0798 + 0.0246i)9-s + (0.224 − 2.99i)10-s + (−0.356 + 1.56i)11-s + (2.55 − 1.00i)12-s + (0.0151 + 0.202i)13-s + (0.776 + 0.117i)14-s + (1.20 − 1.11i)15-s + (−2.99 + 1.44i)16-s + (0.399 + 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.703215 - 0.118981i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.703215 - 0.118981i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (44.7 + 1.84e3i)T \) |
| good | 2 | \( 1 + (5.96 + 4.75i)T + (3.56 + 15.5i)T^{2} \) |
| 3 | \( 1 + (1.39 + 9.26i)T + (-77.4 + 23.8i)T^{2} \) |
| 5 | \( 1 + (-22.1 - 32.5i)T + (-228. + 581. i)T^{2} \) |
| 7 | \( 1 + (17.4 - 10.0i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (43.1 - 189. i)T + (-1.31e4 - 6.35e3i)T^{2} \) |
| 13 | \( 1 + (-2.56 - 34.1i)T + (-2.82e4 + 4.25e3i)T^{2} \) |
| 17 | \( 1 + (-115. - 78.7i)T + (3.05e4 + 7.77e4i)T^{2} \) |
| 19 | \( 1 + (-97.4 + 316. i)T + (-1.07e5 - 7.34e4i)T^{2} \) |
| 23 | \( 1 + (-605. - 562. i)T + (2.09e4 + 2.79e5i)T^{2} \) |
| 29 | \( 1 + (101. - 672. i)T + (-6.75e5 - 2.08e5i)T^{2} \) |
| 31 | \( 1 + (28.0 + 71.4i)T + (-6.76e5 + 6.28e5i)T^{2} \) |
| 37 | \( 1 + (-1.71e3 - 990. i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (464. - 582. i)T + (-6.28e5 - 2.75e6i)T^{2} \) |
| 47 | \( 1 + (405. + 1.77e3i)T + (-4.39e6 + 2.11e6i)T^{2} \) |
| 53 | \( 1 + (182. - 2.43e3i)T + (-7.80e6 - 1.17e6i)T^{2} \) |
| 59 | \( 1 + (-1.09e3 + 527. i)T + (7.55e6 - 9.47e6i)T^{2} \) |
| 61 | \( 1 + (1.25e3 + 493. i)T + (1.01e7 + 9.41e6i)T^{2} \) |
| 67 | \( 1 + (2.70e3 + 833. i)T + (1.66e7 + 1.13e7i)T^{2} \) |
| 71 | \( 1 + (-4.59e3 - 4.95e3i)T + (-1.89e6 + 2.53e7i)T^{2} \) |
| 73 | \( 1 + (446. - 33.4i)T + (2.80e7 - 4.23e6i)T^{2} \) |
| 79 | \( 1 + (1.19e3 + 2.06e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-3.40e3 + 512. i)T + (4.53e7 - 1.39e7i)T^{2} \) |
| 89 | \( 1 + (362. + 2.40e3i)T + (-5.99e7 + 1.84e7i)T^{2} \) |
| 97 | \( 1 + (-682. + 2.98e3i)T + (-7.97e7 - 3.84e7i)T^{2} \) |
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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
−15.26857200397625782395086397226, −13.36788707339907917301850988524, −12.52869266772216886643416466709, −11.34014972042679576565884629403, −10.16674136112281511720010657186, −9.412576909465192899732616926081, −7.46183184332067754808888561186, −6.82126081704748649447789000805, −2.81606954080272634188397853299, −1.64442185381987960159757807304,
0.818388637033611791804145422254, 5.12127229968275561201187481402, 6.07224707830361520842718132568, 8.084089097050103447188554153096, 9.131266248118914624466392978708, 9.875472767423109503868149540293, 10.88056783348839618615217509291, 13.26526748638823487144404173111, 14.60867526941746511294358450023, 16.03304971101039203131840733684
