| L(s) = 1 | + (−0.708 − 1.06i)3-s + (−0.482 + 2.42i)5-s + (2.22 + 11.1i)7-s + (2.82 − 6.81i)9-s + (14.2 + 9.52i)11-s + (0.0771 + 0.0771i)13-s + (2.91 − 1.20i)15-s + (16.9 − 0.824i)17-s + (−3.70 − 8.93i)19-s + (10.2 − 10.2i)21-s + (−25.4 + 38.0i)23-s + (17.4 + 7.22i)25-s + (−20.4 + 4.07i)27-s + (−39.7 − 7.90i)29-s + (27.3 − 18.2i)31-s + ⋯ |
| L(s) = 1 | + (−0.236 − 0.353i)3-s + (−0.0964 + 0.484i)5-s + (0.318 + 1.59i)7-s + (0.313 − 0.757i)9-s + (1.29 + 0.866i)11-s + (0.00593 + 0.00593i)13-s + (0.194 − 0.0803i)15-s + (0.998 − 0.0484i)17-s + (−0.194 − 0.470i)19-s + (0.489 − 0.489i)21-s + (−1.10 + 1.65i)23-s + (0.698 + 0.289i)25-s + (−0.758 + 0.150i)27-s + (−1.37 − 0.272i)29-s + (0.882 − 0.589i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.31340 + 0.456753i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31340 + 0.456753i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (-16.9 + 0.824i)T \) |
| good | 3 | \( 1 + (0.708 + 1.06i)T + (-3.44 + 8.31i)T^{2} \) |
| 5 | \( 1 + (0.482 - 2.42i)T + (-23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (-2.22 - 11.1i)T + (-45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (-14.2 - 9.52i)T + (46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (-0.0771 - 0.0771i)T + 169iT^{2} \) |
| 19 | \( 1 + (3.70 + 8.93i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (25.4 - 38.0i)T + (-202. - 488. i)T^{2} \) |
| 29 | \( 1 + (39.7 + 7.90i)T + (776. + 321. i)T^{2} \) |
| 31 | \( 1 + (-27.3 + 18.2i)T + (367. - 887. i)T^{2} \) |
| 37 | \( 1 + (8.58 + 12.8i)T + (-523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (5.22 + 26.2i)T + (-1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (-9.37 + 22.6i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-5.26 - 5.26i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (22.6 + 54.6i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (40.3 + 16.7i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-93.4 + 18.5i)T + (3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 + 68.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (6.98 + 10.4i)T + (-1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (10.6 - 53.6i)T + (-4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-17.6 - 11.8i)T + (2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (-2.41 + 1.00i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-12.1 + 12.1i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (-160. - 32.0i)T + (8.69e3 + 3.60e3i)T^{2} \) |
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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
−12.78416526639390110204342238289, −11.92726097445391618794937804784, −11.52733590796995847288281685128, −9.737866940330734880640212690628, −9.094036472451350302734791502580, −7.61496999512152935458141433415, −6.49869262949582937539266803392, −5.48033555228573623962318419857, −3.68825332957815987020139128875, −1.86201443819058577040259948978,
1.13293490273352660357547169641, 3.81927186017179173097757372467, 4.68987976812379762202962456805, 6.26808353579361505017102956176, 7.58744992872894756857333330959, 8.568174908877638040117322876145, 10.04950765105459379828667186121, 10.69160453327939634086584305322, 11.74763846289675601268615641302, 12.91539868819164284693225776961
