| L(s) = 1 | + (−1.42 + 1.32i)2-s + (−1.61 − 1.50i)3-s + (0.132 − 1.76i)4-s + (−0.0580 + 0.147i)5-s + 4.28·6-s + (−2.61 + 0.374i)7-s + (−0.273 − 0.342i)8-s + (0.139 + 1.86i)9-s + (−0.112 − 0.287i)10-s + (1.73 − 1.18i)11-s + (−2.86 + 2.66i)12-s + (1.98 + 2.48i)13-s + (3.23 − 3.99i)14-s + (0.315 − 0.152i)15-s + (4.35 + 0.655i)16-s + (1.03 + 2.63i)17-s + ⋯ |
| L(s) = 1 | + (−1.00 + 0.934i)2-s + (−0.933 − 0.866i)3-s + (0.0663 − 0.884i)4-s + (−0.0259 + 0.0660i)5-s + 1.74·6-s + (−0.989 + 0.141i)7-s + (−0.0966 − 0.121i)8-s + (0.0465 + 0.621i)9-s + (−0.0356 − 0.0908i)10-s + (0.524 − 0.357i)11-s + (−0.828 + 0.768i)12-s + (0.550 + 0.689i)13-s + (0.864 − 1.06i)14-s + (0.0814 − 0.0392i)15-s + (1.08 + 0.163i)16-s + (0.250 + 0.639i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.335966 + 0.274167i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.335966 + 0.274167i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.61 - 0.374i)T \) |
| 43 | \( 1 + (-6.46 - 1.07i)T \) |
| good | 2 | \( 1 + (1.42 - 1.32i)T + (0.149 - 1.99i)T^{2} \) |
| 3 | \( 1 + (1.61 + 1.50i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (0.0580 - 0.147i)T + (-3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (-1.73 + 1.18i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (-1.98 - 2.48i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-1.03 - 2.63i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (0.0683 + 0.0465i)T + (6.94 + 17.6i)T^{2} \) |
| 23 | \( 1 + (-0.240 - 3.20i)T + (-22.7 + 3.42i)T^{2} \) |
| 29 | \( 1 + (0.0189 + 0.0829i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-2.66 - 0.822i)T + (25.6 + 17.4i)T^{2} \) |
| 37 | \( 1 + (-5.19 - 8.99i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.703 + 3.08i)T + (-36.9 + 17.7i)T^{2} \) |
| 47 | \( 1 + (7.22 + 4.92i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (0.0552 + 0.140i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (0.525 + 1.34i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (-2.28 + 0.703i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (0.341 - 4.56i)T + (-66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (-10.4 + 5.03i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-6.94 + 1.04i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-5.09 - 8.82i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.89 + 8.29i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (8.33 - 2.57i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + (10.2 + 4.91i)T + (60.4 + 75.8i)T^{2} \) |
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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
−11.93654828240678791348532602570, −11.00076234264314949996421961426, −9.777681921844717190499302219742, −9.012597493571933782233114916003, −7.989369933873725363979939269470, −6.78412591650695111301812802390, −6.51745377049782440145654000042, −5.61550050855060679014976658410, −3.54799135258761008847248331069, −1.11441075758458832040517557066,
0.62533048207159575063353192647, 2.78957758519385701137703001022, 4.14112448573167206785050145499, 5.49932576244558282492534880492, 6.52005036730415154252108036010, 8.031468776472691688569188959230, 9.251767817331853482553098936267, 9.787423774829198291495022235762, 10.60562037406433924350171910857, 11.13312900594306710445055920858
