| L(s) = 1 | + (−1.39 + 0.245i)2-s + (−2.33 − 2.78i)3-s + (1.87 − 0.684i)4-s + (−7.04 − 2.56i)5-s + (3.93 + 3.30i)6-s + (3.79 − 6.57i)7-s + (−2.44 + 1.41i)8-s + (−0.730 + 4.14i)9-s + (10.4 + 1.84i)10-s + (6.18 + 10.7i)11-s + (−6.29 − 3.63i)12-s + (−3.05 + 3.63i)13-s + (−3.67 + 10.0i)14-s + (9.32 + 25.6i)15-s + (3.06 − 2.57i)16-s + (−4.55 − 25.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (−0.778 − 0.928i)3-s + (0.469 − 0.171i)4-s + (−1.40 − 0.513i)5-s + (0.656 + 0.550i)6-s + (0.542 − 0.939i)7-s + (−0.306 + 0.176i)8-s + (−0.0811 + 0.460i)9-s + (1.04 + 0.184i)10-s + (0.562 + 0.973i)11-s + (−0.524 − 0.302i)12-s + (−0.234 + 0.279i)13-s + (−0.262 + 0.721i)14-s + (0.621 + 1.70i)15-s + (0.191 − 0.160i)16-s + (−0.268 − 1.52i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.512 + 0.858i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.227645 - 0.401073i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.227645 - 0.401073i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.245i)T \) |
| 19 | \( 1 + (-10.1 + 16.0i)T \) |
| good | 3 | \( 1 + (2.33 + 2.78i)T + (-1.56 + 8.86i)T^{2} \) |
| 5 | \( 1 + (7.04 + 2.56i)T + (19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (-3.79 + 6.57i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-6.18 - 10.7i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (3.05 - 3.63i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (4.55 + 25.8i)T + (-271. + 98.8i)T^{2} \) |
| 23 | \( 1 + (-26.8 + 9.75i)T + (405. - 340. i)T^{2} \) |
| 29 | \( 1 + (30.4 + 5.36i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (4.94 + 2.85i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 2.36iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-44.0 - 52.5i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-37.3 - 13.5i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-6.27 + 35.5i)T + (-2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + (9.50 + 26.1i)T + (-2.15e3 + 1.80e3i)T^{2} \) |
| 59 | \( 1 + (-39.4 + 6.95i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (20.9 - 7.63i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-18.3 - 3.24i)T + (4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (4.94 - 13.5i)T + (-3.86e3 - 3.24e3i)T^{2} \) |
| 73 | \( 1 + (94.4 - 79.2i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (24.9 + 29.6i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-48.9 + 84.8i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-95.4 + 113. i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-29.4 + 5.19i)T + (8.84e3 - 3.21e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.04777172797065884976294283170, −14.73827369851995257286612955799, −12.97730283972155390199005804230, −11.72904839285407920557783796271, −11.26727878761755062064918407332, −9.247347727523666856877178283999, −7.45997089581062405303613805021, −7.11354970154132432210601988869, −4.63457285618217890268032052273, −0.76090564798095769391046183292,
3.76169009657442672059412132760, 5.74037079343888740415601391184, 7.73040222549583278583696831205, 8.964548058867534271323602787165, 10.73741003580341551303084386561, 11.26559915827561935509319701490, 12.25762992733586188372752237291, 14.79385684001277632759153994209, 15.50505375440150466981984299575, 16.43485112990643150416043131236
