| L(s) = 1 | + (−0.756 − 0.0822i)2-s + (−0.647 + 0.762i)3-s + (−1.38 − 0.305i)4-s + (−0.128 − 0.0978i)5-s + (0.552 − 0.523i)6-s + (1.18 − 2.96i)7-s + (2.46 + 0.830i)8-s + (−0.161 − 0.986i)9-s + (0.0892 + 0.0845i)10-s + (3.21 − 1.48i)11-s + (1.13 − 0.860i)12-s + (0.653 − 3.98i)13-s + (−1.13 + 2.14i)14-s + (0.157 − 0.0347i)15-s + (0.784 + 0.362i)16-s + (0.702 + 1.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.534 − 0.0581i)2-s + (−0.373 + 0.440i)3-s + (−0.694 − 0.152i)4-s + (−0.0575 − 0.0437i)5-s + (0.225 − 0.213i)6-s + (0.447 − 1.12i)7-s + (0.871 + 0.293i)8-s + (−0.0539 − 0.328i)9-s + (0.0282 + 0.0267i)10-s + (0.968 − 0.448i)11-s + (0.326 − 0.248i)12-s + (0.181 − 1.10i)13-s + (−0.304 + 0.573i)14-s + (0.0407 − 0.00897i)15-s + (0.196 + 0.0907i)16-s + (0.170 + 0.427i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.609508 - 0.311154i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.609508 - 0.311154i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.647 - 0.762i)T \) |
| 59 | \( 1 + (7.64 - 0.694i)T \) |
| good | 2 | \( 1 + (0.756 + 0.0822i)T + (1.95 + 0.429i)T^{2} \) |
| 5 | \( 1 + (0.128 + 0.0978i)T + (1.33 + 4.81i)T^{2} \) |
| 7 | \( 1 + (-1.18 + 2.96i)T + (-5.08 - 4.81i)T^{2} \) |
| 11 | \( 1 + (-3.21 + 1.48i)T + (7.12 - 8.38i)T^{2} \) |
| 13 | \( 1 + (-0.653 + 3.98i)T + (-12.3 - 4.15i)T^{2} \) |
| 17 | \( 1 + (-0.702 - 1.76i)T + (-12.3 + 11.6i)T^{2} \) |
| 19 | \( 1 + (2.07 + 3.05i)T + (-7.03 + 17.6i)T^{2} \) |
| 23 | \( 1 + (0.0779 - 1.43i)T + (-22.8 - 2.48i)T^{2} \) |
| 29 | \( 1 + (0.693 - 0.0754i)T + (28.3 - 6.23i)T^{2} \) |
| 31 | \( 1 + (-0.294 + 0.433i)T + (-11.4 - 28.7i)T^{2} \) |
| 37 | \( 1 + (1.54 - 0.522i)T + (29.4 - 22.3i)T^{2} \) |
| 41 | \( 1 + (-0.240 - 4.43i)T + (-40.7 + 4.43i)T^{2} \) |
| 43 | \( 1 + (-6.48 - 3.00i)T + (27.8 + 32.7i)T^{2} \) |
| 47 | \( 1 + (-4.38 + 3.33i)T + (12.5 - 45.2i)T^{2} \) |
| 53 | \( 1 + (5.47 - 5.18i)T + (2.86 - 52.9i)T^{2} \) |
| 61 | \( 1 + (7.22 + 0.785i)T + (59.5 + 13.1i)T^{2} \) |
| 67 | \( 1 + (-14.4 - 4.87i)T + (53.3 + 40.5i)T^{2} \) |
| 71 | \( 1 + (-12.6 + 9.58i)T + (18.9 - 68.4i)T^{2} \) |
| 73 | \( 1 + (4.49 - 8.48i)T + (-40.9 - 60.4i)T^{2} \) |
| 79 | \( 1 + (1.88 + 2.22i)T + (-12.7 + 77.9i)T^{2} \) |
| 83 | \( 1 + (-12.0 + 7.23i)T + (38.8 - 73.3i)T^{2} \) |
| 89 | \( 1 + (8.57 - 0.932i)T + (86.9 - 19.1i)T^{2} \) |
| 97 | \( 1 + (-2.76 - 5.20i)T + (-54.4 + 80.2i)T^{2} \) |
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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
−12.54322028569670899718938320255, −11.12622500152534789603859282856, −10.56968542786746995723825275596, −9.632040844407891273876628524415, −8.567503466848677279646348870310, −7.63129060625691694293331829363, −6.12064965939952553611787691668, −4.73957991518514160697103035927, −3.80472154763217329796391425482, −0.901293179278284022495949381305,
1.76831719726917112588830881620, 4.08614675522171576642157576897, 5.35328327622596500129665922368, 6.67066395670400381310608148343, 7.84030771963657971571030063981, 8.923032115222711053855093898556, 9.489133224540680113504296562666, 10.94798435366321481518455644116, 11.97925199071592079299802149053, 12.56828615568053470100777166893
