| L(s) = 1 | + (−0.415 + 0.909i)2-s + (−0.628 + 0.403i)3-s + (−0.654 − 0.755i)4-s + (−0.142 + 0.989i)5-s + (−0.106 − 0.739i)6-s + (1.26 − 2.76i)7-s + (0.959 − 0.281i)8-s + (−1.01 + 2.22i)9-s + (−0.841 − 0.540i)10-s + (0.533 − 3.70i)11-s + (0.716 + 0.210i)12-s + (0.290 + 0.0852i)13-s + (1.98 + 2.29i)14-s + (−0.310 − 0.679i)15-s + (−0.142 + 0.989i)16-s + (1.20 − 1.39i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (−0.362 + 0.233i)3-s + (−0.327 − 0.377i)4-s + (−0.0636 + 0.442i)5-s + (−0.0434 − 0.301i)6-s + (0.476 − 1.04i)7-s + (0.339 − 0.0996i)8-s + (−0.338 + 0.740i)9-s + (−0.266 − 0.170i)10-s + (0.160 − 1.11i)11-s + (0.206 + 0.0607i)12-s + (0.0805 + 0.0236i)13-s + (0.531 + 0.613i)14-s + (−0.0801 − 0.175i)15-s + (−0.0355 + 0.247i)16-s + (0.292 − 0.337i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04872 + 0.00804469i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04872 + 0.00804469i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 5 | \( 1 + (0.142 - 0.989i)T \) |
| 67 | \( 1 + (5.77 + 5.80i)T \) |
| good | 3 | \( 1 + (0.628 - 0.403i)T + (1.24 - 2.72i)T^{2} \) |
| 7 | \( 1 + (-1.26 + 2.76i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.533 + 3.70i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.290 - 0.0852i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-1.20 + 1.39i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (2.37 + 5.20i)T + (-12.4 + 14.3i)T^{2} \) |
| 23 | \( 1 + (-0.632 + 0.406i)T + (9.55 - 20.9i)T^{2} \) |
| 29 | \( 1 - 7.59T + 29T^{2} \) |
| 31 | \( 1 + (-8.69 + 2.55i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 - 0.755T + 37T^{2} \) |
| 41 | \( 1 + (-1.04 + 1.20i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (6.67 - 7.70i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (-1.73 + 1.11i)T + (19.5 - 42.7i)T^{2} \) |
| 53 | \( 1 + (-5.65 - 6.52i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-13.9 + 4.09i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (1.21 + 8.45i)T + (-58.5 + 17.1i)T^{2} \) |
| 71 | \( 1 + (0.263 + 0.304i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.0846 + 0.588i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-4.91 - 1.44i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-0.0360 + 0.250i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-3.52 - 2.26i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−10.63384711095904894748785785288, −9.743061830886693691436178508636, −8.482597957636158815262978190938, −7.986821144771589841400031202171, −6.92408324674308789787399253351, −6.19875932760150280382184484265, −5.05215532334065329271019853179, −4.27610697252106894282882172909, −2.79657284148781529828720369071, −0.75776017695019758193404920555,
1.28746631125813298142726579185, 2.48634889443594851147283225640, 3.90483230267354343140371298390, 4.98852751991227709978533612186, 5.94211117860040307845846217126, 6.97430037314326345074429145676, 8.343066087397646674949074885632, 8.640489823391870363690503396187, 9.778136648791333312496636382968, 10.38874067875235449609397588206
