| L(s) = 1 | + (−0.0268 − 0.00282i)3-s + (−4.16 + 7.20i)5-s + (0.359 − 0.0763i)7-s + (−8.80 − 1.87i)9-s + (−7.33 + 6.60i)11-s + (6.15 + 13.8i)13-s + (0.132 − 0.181i)15-s + (−3.02 − 2.72i)17-s + (27.6 + 12.3i)19-s + (−0.00985 + 0.00103i)21-s + (−8.65 + 2.81i)23-s + (−22.1 − 38.3i)25-s + (0.462 + 0.150i)27-s + (−15.4 − 21.2i)29-s + (9.16 + 29.6i)31-s + ⋯ |
| L(s) = 1 | + (−0.00895 − 0.000941i)3-s + (−0.832 + 1.44i)5-s + (0.0512 − 0.0109i)7-s + (−0.978 − 0.207i)9-s + (−0.667 + 0.600i)11-s + (0.473 + 1.06i)13-s + (0.00880 − 0.0121i)15-s + (−0.177 − 0.160i)17-s + (1.45 + 0.648i)19-s + (−0.000469 + 4.93e−5i)21-s + (−0.376 + 0.122i)23-s + (−0.885 − 1.53i)25-s + (0.0171 + 0.00556i)27-s + (−0.531 − 0.731i)29-s + (0.295 + 0.955i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.441421 + 0.749670i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.441421 + 0.749670i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (-9.16 - 29.6i)T \) |
| good | 3 | \( 1 + (0.0268 + 0.00282i)T + (8.80 + 1.87i)T^{2} \) |
| 5 | \( 1 + (4.16 - 7.20i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-0.359 + 0.0763i)T + (44.7 - 19.9i)T^{2} \) |
| 11 | \( 1 + (7.33 - 6.60i)T + (12.6 - 120. i)T^{2} \) |
| 13 | \( 1 + (-6.15 - 13.8i)T + (-113. + 125. i)T^{2} \) |
| 17 | \( 1 + (3.02 + 2.72i)T + (30.2 + 287. i)T^{2} \) |
| 19 | \( 1 + (-27.6 - 12.3i)T + (241. + 268. i)T^{2} \) |
| 23 | \( 1 + (8.65 - 2.81i)T + (427. - 310. i)T^{2} \) |
| 29 | \( 1 + (15.4 + 21.2i)T + (-259. + 799. i)T^{2} \) |
| 37 | \( 1 + (-44.9 + 25.9i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (3.28 + 31.2i)T + (-1.64e3 + 349. i)T^{2} \) |
| 43 | \( 1 + (-20.2 + 45.5i)T + (-1.23e3 - 1.37e3i)T^{2} \) |
| 47 | \( 1 + (-49.5 - 36.0i)T + (682. + 2.10e3i)T^{2} \) |
| 53 | \( 1 + (12.9 - 61.1i)T + (-2.56e3 - 1.14e3i)T^{2} \) |
| 59 | \( 1 + (6.01 - 57.2i)T + (-3.40e3 - 723. i)T^{2} \) |
| 61 | \( 1 + 17.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (11.9 - 20.7i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-101. - 21.6i)T + (4.60e3 + 2.05e3i)T^{2} \) |
| 73 | \( 1 + (80.1 - 72.1i)T + (557. - 5.29e3i)T^{2} \) |
| 79 | \( 1 + (-19.3 - 17.4i)T + (652. + 6.20e3i)T^{2} \) |
| 83 | \( 1 + (51.4 - 5.40i)T + (6.73e3 - 1.43e3i)T^{2} \) |
| 89 | \( 1 + (-68.1 - 22.1i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (25.6 - 78.8i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−13.88521095227909082316607019921, −12.15472214051098237935746025838, −11.43454732330627406101345604008, −10.62619423094872728117815076123, −9.358749368960280313862926184300, −7.906196040585051724871702080768, −7.08071567926931047877140168928, −5.81890522839281463383891820074, −3.96379533561006363015221847420, −2.69557830859502147437159875027,
0.60506484783844047744088195702, 3.22256057476395133014642559037, 4.87259913905983471432691434868, 5.76595008542512457023331198280, 7.87281365009285634471269115759, 8.319128504752243830963657994125, 9.464481386194426466916746970245, 11.05289682371642403533179629368, 11.74128231683955669167359606932, 12.92658732861824583184263588092
