| L(s) = 1 | + (−0.142 + 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (−0.654 − 0.755i)5-s + (−0.959 + 0.281i)6-s + (−1.19 − 0.766i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (0.841 − 0.540i)10-s + (0.577 + 4.01i)11-s + (−0.142 − 0.989i)12-s + (−3.17 + 2.04i)13-s + (0.928 − 1.07i)14-s + (0.415 − 0.909i)15-s + (0.841 + 0.540i)16-s + (−0.685 + 0.201i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (0.239 + 0.525i)3-s + (−0.479 − 0.140i)4-s + (−0.292 − 0.337i)5-s + (−0.391 + 0.115i)6-s + (−0.450 − 0.289i)7-s + (0.146 − 0.321i)8-s + (−0.218 + 0.251i)9-s + (0.266 − 0.170i)10-s + (0.174 + 1.21i)11-s + (−0.0410 − 0.285i)12-s + (−0.881 + 0.566i)13-s + (0.248 − 0.286i)14-s + (0.107 − 0.234i)15-s + (0.210 + 0.135i)16-s + (−0.166 + 0.0487i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.110352 - 0.416309i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.110352 - 0.416309i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 3 | \( 1 + (-0.415 - 0.909i)T \) |
| 5 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (1.59 + 4.52i)T \) |
| good | 7 | \( 1 + (1.19 + 0.766i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.577 - 4.01i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (3.17 - 2.04i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.685 - 0.201i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (5.06 + 1.48i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (6.08 - 1.78i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (4.36 - 9.55i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-1.24 + 1.44i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (7.44 + 8.58i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (0.495 + 1.08i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 6.23T + 47T^{2} \) |
| 53 | \( 1 + (-10.2 - 6.60i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-10.2 + 6.56i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (4.37 - 9.57i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (1.53 - 10.7i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (1.76 - 12.2i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (2.26 + 0.665i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (6.51 - 4.18i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-11.2 + 12.9i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (3.42 + 7.49i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-1.48 - 1.71i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.58230478805963084878754738465, −10.04897878945333344044508487866, −9.039354446453431217645134982933, −8.603382037583922232845575856973, −7.18085259120776832836453637533, −6.94970156113424037080676814958, −5.47423425000517028522944163127, −4.53608077054231140224034343326, −3.87586313806253286890119588843, −2.16661228530520108661590639939,
0.21355185442609011091622022329, 2.06567229227741905572060265074, 3.11227248341695048528230825235, 3.98664118266589795189284609100, 5.53115616445841547652046196502, 6.35586920714812456169823032055, 7.57962215912170552312111853284, 8.226005849500184628511066214843, 9.207442955758487553353616379884, 9.946111945901501127639033463871
