| L(s) = 1 | + (0.167 + 1.05i)2-s + (−1.54 − 0.786i)3-s + (2.71 − 0.880i)4-s + (4.35 − 2.44i)5-s + (0.573 − 1.76i)6-s + (−2.58 − 2.58i)7-s + (3.33 + 6.54i)8-s + (1.76 + 2.42i)9-s + (3.32 + 4.20i)10-s + (12.9 + 9.40i)11-s + (−4.87 − 0.772i)12-s + (1.48 − 9.38i)13-s + (2.30 − 3.17i)14-s + (−8.65 + 0.351i)15-s + (2.85 − 2.07i)16-s + (−18.8 + 9.62i)17-s + ⋯ |
| L(s) = 1 | + (0.0838 + 0.529i)2-s + (−0.514 − 0.262i)3-s + (0.677 − 0.220i)4-s + (0.871 − 0.489i)5-s + (0.0956 − 0.294i)6-s + (−0.369 − 0.369i)7-s + (0.416 + 0.817i)8-s + (0.195 + 0.269i)9-s + (0.332 + 0.420i)10-s + (1.17 + 0.855i)11-s + (−0.406 − 0.0643i)12-s + (0.114 − 0.722i)13-s + (0.164 − 0.226i)14-s + (−0.576 + 0.0234i)15-s + (0.178 − 0.129i)16-s + (−1.11 + 0.565i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.44128 + 0.0812177i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44128 + 0.0812177i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.54 + 0.786i)T \) |
| 5 | \( 1 + (-4.35 + 2.44i)T \) |
| good | 2 | \( 1 + (-0.167 - 1.05i)T + (-3.80 + 1.23i)T^{2} \) |
| 7 | \( 1 + (2.58 + 2.58i)T + 49iT^{2} \) |
| 11 | \( 1 + (-12.9 - 9.40i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-1.48 + 9.38i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (18.8 - 9.62i)T + (169. - 233. i)T^{2} \) |
| 19 | \( 1 + (30.6 + 9.96i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (20.7 - 3.29i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (-22.9 + 7.45i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (14.1 - 43.4i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (-10.7 - 1.69i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (18.1 - 13.2i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (32.1 - 32.1i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (17.0 - 33.4i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-5.72 - 2.91i)T + (1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (36.9 + 50.9i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-40.7 - 29.5i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-59.0 + 30.0i)T + (2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (14.3 + 44.1i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-22.2 + 3.51i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-44.5 + 14.4i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (15.9 + 31.3i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (62.8 - 86.5i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-78.4 + 153. i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−14.41089816652974502436187488094, −13.19523772689502701410461929210, −12.28937852370538410911514888207, −10.92603506288287304913043043751, −9.970059148535816098760669143780, −8.455717560850412874145263811331, −6.73120703706449455224035926672, −6.26191386346881621240013145524, −4.69234418916737724042586812029, −1.84466245248328363735509390898,
2.15388798062039248561702386183, 3.93126508948230564210017285185, 6.16764238759399987127460140460, 6.67023560231235083792070270836, 8.877950348368806672837788784817, 10.05896075884740021212831373617, 11.09066499144704491466449583935, 11.83174227395510196413364457082, 13.03634388394906142770292821727, 14.17927515554015906494218907629
