| 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
−12.24633526700994671312137274732, −11.06639991111835309261898902052, −9.801822969156213362105273756195, −9.158696551181339765620589583873, −8.370297439866376956095644266180, −6.93261514480669055519107248355, −6.10093465894330274300424348139, −5.49665518360399430979286114059, −2.48903584032105256878654876485, −1.23587853931797700042451828096,
2.20754430004637706206690872151, 3.30669181518177960696055978308, 5.18963685833852583976346530160, 6.40982179825645419856015150606, 7.51259121123617063451463622103, 9.250675786245978540246151569919, 9.696329659420198585024845934658, 10.15266345520366495595005057226, 11.04846187156975270394422306999, 12.38062823444956926430754233798
