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
−9.946111945901501127639033463871, −9.207442955758487553353616379884, −8.226005849500184628511066214843, −7.57962215912170552312111853284, −6.35586920714812456169823032055, −5.53115616445841547652046196502, −3.98664118266589795189284609100, −3.11227248341695048528230825235, −2.06567229227741905572060265074, −0.21355185442609011091622022329,
2.16661228530520108661590639939, 3.87586313806253286890119588843, 4.53608077054231140224034343326, 5.47423425000517028522944163127, 6.94970156113424037080676814958, 7.18085259120776832836453637533, 8.603382037583922232845575856973, 9.039354446453431217645134982933, 10.04897878945333344044508487866, 10.58230478805963084878754738465