| L(s) = 1 | + (−0.173 − 0.984i)2-s + (−1.38 − 1.04i)3-s + (−0.939 + 0.342i)4-s + (2.46 + 2.07i)5-s + (−0.791 + 1.54i)6-s + (−0.939 − 0.342i)7-s + (0.5 + 0.866i)8-s + (0.809 + 2.88i)9-s + (1.61 − 2.79i)10-s + (−4.20 + 3.53i)11-s + (1.65 + 0.511i)12-s + (−1.05 + 5.96i)13-s + (−0.173 + 0.984i)14-s + (−1.23 − 5.44i)15-s + (0.766 − 0.642i)16-s + (0.660 − 1.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.796 − 0.604i)3-s + (−0.469 + 0.171i)4-s + (1.10 + 0.926i)5-s + (−0.322 + 0.629i)6-s + (−0.355 − 0.129i)7-s + (0.176 + 0.306i)8-s + (0.269 + 0.962i)9-s + (0.509 − 0.882i)10-s + (−1.26 + 1.06i)11-s + (0.477 + 0.147i)12-s + (−0.291 + 1.65i)13-s + (−0.0464 + 0.263i)14-s + (−0.319 − 1.40i)15-s + (0.191 − 0.160i)16-s + (0.160 − 0.277i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.526i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.849 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.828686 + 0.236061i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.828686 + 0.236061i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 3 | \( 1 + (1.38 + 1.04i)T \) |
| 7 | \( 1 + (0.939 + 0.342i)T \) |
| good | 5 | \( 1 + (-2.46 - 2.07i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (4.20 - 3.53i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (1.05 - 5.96i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.660 + 1.14i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.742 - 1.28i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.21 + 2.26i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.333 - 1.89i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.42 - 0.880i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (3.77 - 6.52i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.76 + 10.0i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (1.25 - 1.05i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-5.91 - 2.15i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 7.02T + 53T^{2} \) |
| 59 | \( 1 + (3.81 + 3.20i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (4.74 + 1.72i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.72 - 9.80i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (7.05 - 12.2i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.766 - 1.32i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.871 - 4.94i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (2.20 + 12.5i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (5.14 + 8.91i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.186 + 0.156i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.36922299218854415528280542319, −10.43099868127565729560354816895, −10.06332432910895067753763450069, −8.978879041266639352501738405786, −7.27594135290096818748890895368, −6.88163372764422363570405384760, −5.62764618151243360079452516358, −4.63355563426410128699816124257, −2.70840639123037717817772745654, −1.82406460799633166186341270310,
0.66445127373209163102491517084, 3.13959757689274662612384853078, 4.94321776972132758560975720591, 5.53890113627113880424671687028, 6.00873705914554787302384458957, 7.52870906519737483916023706477, 8.662169886225240234091968154070, 9.437485366427333152517914158111, 10.29795305191150417299416968801, 10.92888818449955381835170149955
