| L(s) = 1 | + (−0.777 + 1.18i)2-s + (−3.02 − 0.524i)3-s + (−0.791 − 1.83i)4-s + (−0.967 + 2.92i)5-s + (2.97 − 3.16i)6-s + (0.196 − 4.00i)7-s + (2.78 + 0.493i)8-s + (6.05 + 2.16i)9-s + (−2.70 − 3.42i)10-s + (−3.73 − 1.65i)11-s + (1.42 + 5.97i)12-s + (1.89 − 1.63i)13-s + (4.57 + 3.34i)14-s + (4.46 − 8.35i)15-s + (−2.74 + 2.90i)16-s + (2.32 − 1.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.549 + 0.835i)2-s + (−1.74 − 0.303i)3-s + (−0.395 − 0.918i)4-s + (−0.432 + 1.30i)5-s + (1.21 − 1.29i)6-s + (0.0743 − 1.51i)7-s + (0.984 + 0.174i)8-s + (2.01 + 0.721i)9-s + (−0.856 − 1.08i)10-s + (−1.12 − 0.499i)11-s + (0.412 + 1.72i)12-s + (0.524 − 0.452i)13-s + (1.22 + 0.894i)14-s + (1.15 − 2.15i)15-s + (−0.687 + 0.726i)16-s + (0.564 − 0.301i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.536 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.148812 + 0.270859i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.148812 + 0.270859i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.777 - 1.18i)T \) |
| good | 3 | \( 1 + (3.02 + 0.524i)T + (2.82 + 1.01i)T^{2} \) |
| 5 | \( 1 + (0.967 - 2.92i)T + (-4.01 - 2.97i)T^{2} \) |
| 7 | \( 1 + (-0.196 + 4.00i)T + (-6.96 - 0.686i)T^{2} \) |
| 11 | \( 1 + (3.73 + 1.65i)T + (7.38 + 8.15i)T^{2} \) |
| 13 | \( 1 + (-1.89 + 1.63i)T + (1.90 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.32 + 1.24i)T + (9.44 - 14.1i)T^{2} \) |
| 19 | \( 1 + (0.0744 - 0.0421i)T + (9.76 - 16.2i)T^{2} \) |
| 23 | \( 1 + (-1.71 - 6.86i)T + (-20.2 + 10.8i)T^{2} \) |
| 29 | \( 1 + (6.06 - 0.148i)T + (28.9 - 1.42i)T^{2} \) |
| 31 | \( 1 + (-0.434 + 0.290i)T + (11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (3.63 + 0.448i)T + (35.8 + 8.99i)T^{2} \) |
| 41 | \( 1 + (4.93 - 3.65i)T + (11.9 - 39.2i)T^{2} \) |
| 43 | \( 1 + (4.92 - 6.99i)T + (-14.4 - 40.4i)T^{2} \) |
| 47 | \( 1 + (-1.17 - 11.9i)T + (-46.0 + 9.16i)T^{2} \) |
| 53 | \( 1 + (-12.3 - 0.302i)T + (52.9 + 2.60i)T^{2} \) |
| 59 | \( 1 + (1.40 - 1.63i)T + (-8.65 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-7.97 + 1.78i)T + (55.1 - 26.0i)T^{2} \) |
| 67 | \( 1 + (-12.6 - 8.03i)T + (28.6 + 60.5i)T^{2} \) |
| 71 | \( 1 + (3.53 + 1.66i)T + (45.0 + 54.8i)T^{2} \) |
| 73 | \( 1 + (0.170 + 3.46i)T + (-72.6 + 7.15i)T^{2} \) |
| 79 | \( 1 + (0.300 + 0.365i)T + (-15.4 + 77.4i)T^{2} \) |
| 83 | \( 1 + (-8.57 + 1.05i)T + (80.5 - 20.1i)T^{2} \) |
| 89 | \( 1 + (-1.08 + 4.31i)T + (-78.4 - 41.9i)T^{2} \) |
| 97 | \( 1 + (3.48 - 17.5i)T + (-89.6 - 37.1i)T^{2} \) |
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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
−11.01186204471584355522148262170, −10.49151338294594733028978700624, −9.880743934040186673538270781598, −7.87706435298219689596730537708, −7.40491043891991949332190484862, −6.77106982564428179109029394390, −5.83852791433658820549614786761, −5.03974966590831443825867254492, −3.63973915039997717520312430938, −1.01252977878011207127237393424,
0.35947571386502031089701353050, 1.97637101142049619521337356988, 3.97503897218055847299286439084, 5.13573529614060186209119770792, 5.38490527320878636746256915986, 6.90703391510826109808553910398, 8.324596071609539948927458986762, 8.858944303493108193410982978053, 9.950875616412486363165421081237, 10.67277181811133937797017212775
