L(s) = 1 | + (0.5 − 0.866i)2-s + (−2.50 + 1.44i)3-s + (−0.499 − 0.866i)4-s + (1.41 + 1.73i)5-s + 2.88i·6-s + (−0.668 + 0.385i)7-s − 0.999·8-s + (2.67 − 4.63i)9-s + (2.20 − 0.359i)10-s − 4.03·11-s + (2.50 + 1.44i)12-s + (−1.92 − 3.32i)13-s + 0.771i·14-s + (−6.04 − 2.28i)15-s + (−0.5 + 0.866i)16-s + (−3.69 + 6.39i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−1.44 + 0.834i)3-s + (−0.249 − 0.433i)4-s + (0.632 + 0.774i)5-s + 1.17i·6-s + (−0.252 + 0.145i)7-s − 0.353·8-s + (0.891 − 1.54i)9-s + (0.697 − 0.113i)10-s − 1.21·11-s + (0.722 + 0.417i)12-s + (−0.532 − 0.922i)13-s + 0.206i·14-s + (−1.56 − 0.590i)15-s + (−0.125 + 0.216i)16-s + (−0.895 + 1.55i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 - 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0827431 + 0.299467i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0827431 + 0.299467i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (-1.41 - 1.73i)T \) |
| 37 | \( 1 + (-4.87 + 3.63i)T \) |
good | 3 | \( 1 + (2.50 - 1.44i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (0.668 - 0.385i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 4.03T + 11T^{2} \) |
| 13 | \( 1 + (1.92 + 3.32i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.69 - 6.39i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.31 - 1.33i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 4.28T + 23T^{2} \) |
| 29 | \( 1 + 3.03iT - 29T^{2} \) |
| 31 | \( 1 - 0.197iT - 31T^{2} \) |
| 41 | \( 1 + (-2.56 - 4.44i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 + 1.03iT - 47T^{2} \) |
| 53 | \( 1 + (9.81 + 5.66i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.08 - 3.51i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.52 + 3.19i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.06 + 2.92i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.15 - 12.3i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 3.22iT - 73T^{2} \) |
| 79 | \( 1 + (6.11 - 3.52i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-12.9 - 7.48i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.24 + 3.60i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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
−11.44319646483017445649111458803, −10.82483736491195098078587178404, −10.18613233580175343059411601925, −9.773792884596424252724157623073, −8.091161871899991116519260614328, −6.43429723158763503733089104249, −5.86049818974190486657408483573, −5.00098468109429303654187780910, −3.81846134753563128350107995172, −2.36355127407092143965469883608,
0.20560568494274588677673347457, 2.21605536801492708899997198469, 4.70283090696739530253431749075, 5.14106387977796633619187787193, 6.23735881307985453530157147000, 6.89578521037142659447243654019, 7.86341812053246283361409048011, 9.138719205629409548497475523245, 10.16890003109793944938562250289, 11.32744666641781234390580897650