| L(s) = 1 | + (1.06 − 0.933i)2-s + (−0.241 − 0.361i)3-s + (0.257 − 1.98i)4-s + (−1.43 − 0.286i)5-s + (−0.594 − 0.158i)6-s + (1.35 − 0.270i)7-s + (−1.57 − 2.34i)8-s + (1.07 − 2.59i)9-s + (−1.79 + 1.03i)10-s + (4.33 + 2.89i)11-s + (−0.780 + 0.386i)12-s + (−4.55 + 4.55i)13-s + (1.19 − 1.55i)14-s + (0.244 + 0.589i)15-s + (−3.86 − 1.02i)16-s + (0.599 + 4.07i)17-s + ⋯ |
| L(s) = 1 | + (0.751 − 0.659i)2-s + (−0.139 − 0.208i)3-s + (0.128 − 0.991i)4-s + (−0.643 − 0.127i)5-s + (−0.242 − 0.0648i)6-s + (0.513 − 0.102i)7-s + (−0.557 − 0.830i)8-s + (0.358 − 0.865i)9-s + (−0.567 + 0.328i)10-s + (1.30 + 0.874i)11-s + (−0.225 + 0.111i)12-s + (−1.26 + 1.26i)13-s + (0.318 − 0.415i)14-s + (0.0630 + 0.152i)15-s + (−0.966 − 0.255i)16-s + (0.145 + 0.989i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11341 - 0.928944i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11341 - 0.928944i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.06 + 0.933i)T \) |
| 17 | \( 1 + (-0.599 - 4.07i)T \) |
| good | 3 | \( 1 + (0.241 + 0.361i)T + (-1.14 + 2.77i)T^{2} \) |
| 5 | \( 1 + (1.43 + 0.286i)T + (4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (-1.35 + 0.270i)T + (6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-4.33 - 2.89i)T + (4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (4.55 - 4.55i)T - 13iT^{2} \) |
| 19 | \( 1 + (0.0890 + 0.215i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-5.20 - 3.47i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-1.29 + 6.52i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (-0.132 - 0.198i)T + (-11.8 + 28.6i)T^{2} \) |
| 37 | \( 1 + (2.30 - 1.54i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (0.387 + 1.94i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (1.82 - 4.39i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-5.81 + 5.81i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.13 + 1.29i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (7.79 + 3.22i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (0.796 + 4.00i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + 8.12iT - 67T^{2} \) |
| 71 | \( 1 + (12.8 - 8.60i)T + (27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (1.07 - 5.41i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-0.800 + 1.19i)T + (-30.2 - 72.9i)T^{2} \) |
| 83 | \( 1 + (-9.56 + 3.96i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (0.0723 - 0.0723i)T - 89iT^{2} \) |
| 97 | \( 1 + (8.14 + 1.62i)T + (89.6 + 37.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.67165011955884632313233891711, −11.91184855487503089405959951027, −11.53062151168951181189703895885, −9.947818645883953626512650104745, −9.149611340609410587676475771555, −7.31037210773684934690821068905, −6.39835354788213391137074354575, −4.63923975119094890529974850086, −3.87770133882136361890820812451, −1.68954228492837557270597014564,
3.07126520377543563117536674989, 4.55338577531950605996134227965, 5.46369593055052373942781426663, 7.05365348779574001698655326120, 7.83534976976275748282129906152, 8.989084815609680173368356536593, 10.66792265158579450868580681923, 11.62846107840223926649431437952, 12.43465072036847868956095174982, 13.62063648893702109880843922185
