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 + (−0.306 + 0.197i)7-s + (−0.415 − 0.909i)8-s + (−0.654 − 0.755i)9-s + (−0.841 − 0.540i)10-s + (−0.338 + 2.35i)11-s + (0.142 − 0.989i)12-s + (−2.18 − 1.40i)13-s + (−0.238 − 0.275i)14-s + (−0.415 − 0.909i)15-s + (0.841 − 0.540i)16-s + (−5.18 − 1.52i)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 + (−0.115 + 0.0745i)7-s + (−0.146 − 0.321i)8-s + (−0.218 − 0.251i)9-s + (−0.266 − 0.170i)10-s + (−0.102 + 0.709i)11-s + (0.0410 − 0.285i)12-s + (−0.607 − 0.390i)13-s + (−0.0638 − 0.0736i)14-s + (−0.107 − 0.234i)15-s + (0.210 − 0.135i)16-s + (−1.25 − 0.369i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 + 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.146763 - 0.247633i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.146763 - 0.247633i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-4.55 - 1.51i)T \) |
good | 7 | \( 1 + (0.306 - 0.197i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.338 - 2.35i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (2.18 + 1.40i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (5.18 + 1.52i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (4.82 - 1.41i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (0.347 + 0.102i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (2.23 + 4.89i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (4.14 + 4.78i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (3.08 - 3.55i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.74 + 6.00i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 5.31T + 47T^{2} \) |
| 53 | \( 1 + (-7.10 + 4.56i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-2.37 - 1.52i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-4.31 - 9.45i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (1.04 + 7.24i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.79 - 12.5i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (12.7 - 3.74i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (9.84 + 6.32i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-0.520 - 0.600i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (5.16 - 11.3i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (4.79 - 5.53i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.92593832448234498286657533795, −10.14039192828876759494093779260, −9.266911639347764157414199120958, −8.470373630076441204851144519299, −7.34710657230433335819469551320, −6.74795332033550614075604072645, −5.64336777713760443698300694921, −4.71171177094596892077570572573, −3.89034551977237575120360483129, −2.47379651176377761514388443664,
0.14431813498502792761097024116, 1.77524850491333608379608414878, 3.00350209828709948829567057353, 4.30269307758197808452324615533, 5.13285679402322228563968146539, 6.37718612290404247560554375777, 7.14812378432786595415530614002, 8.528504792362677093057907346514, 8.807602021079193645399875347349, 10.08730869859084575640156969319