| L(s) = 1 | + (−1.26 − 0.635i)2-s + (0.0447 + 1.73i)3-s + (1.19 + 1.60i)4-s + (4.00 + 1.45i)5-s + (1.04 − 2.21i)6-s + (−1.46 − 0.258i)7-s + (−0.487 − 2.78i)8-s + (−2.99 + 0.154i)9-s + (−4.13 − 4.38i)10-s + (1.57 + 4.33i)11-s + (−2.72 + 2.13i)12-s + (−2.82 − 3.36i)13-s + (1.68 + 1.25i)14-s + (−2.34 + 6.99i)15-s + (−1.15 + 3.83i)16-s + (2.43 − 1.40i)17-s + ⋯ |
| L(s) = 1 | + (−0.893 − 0.449i)2-s + (0.0258 + 0.999i)3-s + (0.596 + 0.802i)4-s + (1.79 + 0.651i)5-s + (0.425 − 0.904i)6-s + (−0.554 − 0.0977i)7-s + (−0.172 − 0.985i)8-s + (−0.998 + 0.0516i)9-s + (−1.30 − 1.38i)10-s + (0.475 + 1.30i)11-s + (−0.786 + 0.617i)12-s + (−0.784 − 0.934i)13-s + (0.451 + 0.336i)14-s + (−0.605 + 1.80i)15-s + (−0.288 + 0.957i)16-s + (0.590 − 0.340i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.855039 + 0.504036i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.855039 + 0.504036i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.26 + 0.635i)T \) |
| 3 | \( 1 + (-0.0447 - 1.73i)T \) |
| good | 5 | \( 1 + (-4.00 - 1.45i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.46 + 0.258i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-1.57 - 4.33i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (2.82 + 3.36i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.43 + 1.40i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.516 + 0.895i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.345 - 1.95i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.783 + 0.657i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (6.04 - 1.06i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-5.15 + 2.97i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.77 - 3.30i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.28 - 0.830i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.806 + 4.57i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 3.14T + 53T^{2} \) |
| 59 | \( 1 + (-2.77 + 7.62i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (2.25 + 0.397i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.24 + 6.07i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.21 + 3.84i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.12 + 10.6i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.19 + 2.62i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.23 + 3.85i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (3.14 + 1.81i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.20 - 1.89i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.38062823444956926430754233798, −11.04846187156975270394422306999, −10.15266345520366495595005057226, −9.696329659420198585024845934658, −9.250675786245978540246151569919, −7.51259121123617063451463622103, −6.40982179825645419856015150606, −5.18963685833852583976346530160, −3.30669181518177960696055978308, −2.20754430004637706206690872151,
1.23587853931797700042451828096, 2.48903584032105256878654876485, 5.49665518360399430979286114059, 6.10093465894330274300424348139, 6.93261514480669055519107248355, 8.370297439866376956095644266180, 9.158696551181339765620589583873, 9.801822969156213362105273756195, 11.06639991111835309261898902052, 12.24633526700994671312137274732
