| L(s) = 1 | + (0.283 + 0.0308i)2-s + (0.647 − 0.762i)3-s + (−1.87 − 0.412i)4-s + (2.97 + 2.26i)5-s + (0.207 − 0.196i)6-s + (1.21 − 3.03i)7-s + (−1.05 − 0.357i)8-s + (−0.161 − 0.986i)9-s + (0.774 + 0.733i)10-s + (2.37 − 1.09i)11-s + (−1.52 + 1.16i)12-s + (−0.960 + 5.85i)13-s + (0.437 − 0.825i)14-s + (3.65 − 0.803i)15-s + (3.19 + 1.47i)16-s + (−1.39 − 3.50i)17-s + ⋯ |
| L(s) = 1 | + (0.200 + 0.0218i)2-s + (0.373 − 0.440i)3-s + (−0.936 − 0.206i)4-s + (1.33 + 1.01i)5-s + (0.0845 − 0.0801i)6-s + (0.457 − 1.14i)7-s + (−0.374 − 0.126i)8-s + (−0.0539 − 0.328i)9-s + (0.244 + 0.232i)10-s + (0.714 − 0.330i)11-s + (−0.440 + 0.335i)12-s + (−0.266 + 1.62i)13-s + (0.116 − 0.220i)14-s + (0.942 − 0.207i)15-s + (0.798 + 0.369i)16-s + (−0.338 − 0.850i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.42348 - 0.233189i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42348 - 0.233189i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-5.82 + 5.00i)T \) |
| good | 2 | \( 1 + (-0.283 - 0.0308i)T + (1.95 + 0.429i)T^{2} \) |
| 5 | \( 1 + (-2.97 - 2.26i)T + (1.33 + 4.81i)T^{2} \) |
| 7 | \( 1 + (-1.21 + 3.03i)T + (-5.08 - 4.81i)T^{2} \) |
| 11 | \( 1 + (-2.37 + 1.09i)T + (7.12 - 8.38i)T^{2} \) |
| 13 | \( 1 + (0.960 - 5.85i)T + (-12.3 - 4.15i)T^{2} \) |
| 17 | \( 1 + (1.39 + 3.50i)T + (-12.3 + 11.6i)T^{2} \) |
| 19 | \( 1 + (2.54 + 3.75i)T + (-7.03 + 17.6i)T^{2} \) |
| 23 | \( 1 + (0.0711 - 1.31i)T + (-22.8 - 2.48i)T^{2} \) |
| 29 | \( 1 + (6.69 - 0.727i)T + (28.3 - 6.23i)T^{2} \) |
| 31 | \( 1 + (4.50 - 6.64i)T + (-11.4 - 28.7i)T^{2} \) |
| 37 | \( 1 + (8.46 - 2.85i)T + (29.4 - 22.3i)T^{2} \) |
| 41 | \( 1 + (-0.164 - 3.03i)T + (-40.7 + 4.43i)T^{2} \) |
| 43 | \( 1 + (1.77 + 0.819i)T + (27.8 + 32.7i)T^{2} \) |
| 47 | \( 1 + (3.24 - 2.47i)T + (12.5 - 45.2i)T^{2} \) |
| 53 | \( 1 + (-8.29 + 7.86i)T + (2.86 - 52.9i)T^{2} \) |
| 61 | \( 1 + (-0.794 - 0.0864i)T + (59.5 + 13.1i)T^{2} \) |
| 67 | \( 1 + (-1.93 - 0.652i)T + (53.3 + 40.5i)T^{2} \) |
| 71 | \( 1 + (4.18 - 3.18i)T + (18.9 - 68.4i)T^{2} \) |
| 73 | \( 1 + (-5.35 + 10.0i)T + (-40.9 - 60.4i)T^{2} \) |
| 79 | \( 1 + (0.599 + 0.705i)T + (-12.7 + 77.9i)T^{2} \) |
| 83 | \( 1 + (-12.7 + 7.64i)T + (38.8 - 73.3i)T^{2} \) |
| 89 | \( 1 + (-7.65 + 0.832i)T + (86.9 - 19.1i)T^{2} \) |
| 97 | \( 1 + (-1.02 - 1.93i)T + (-54.4 + 80.2i)T^{2} \) |
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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
−13.19825305172308401774047064243, −11.57523046370341652241813631571, −10.57325834734266208942467170919, −9.501254412180072599135664764562, −8.898395466549078716747299990083, −7.10788183815377902774923222122, −6.55240580629624789216066272228, −5.00215324750031787059913575402, −3.64265271440815510259754144771, −1.79678421855175831129092566983,
2.07187815996240196137592839414, 3.97665538538727659986399787529, 5.36634995885827584788640634797, 5.70980331632147826451486480539, 8.145425582494467531963657636961, 8.830672448304901946684860560517, 9.503658855859319401879885151036, 10.43381618346044486968907339864, 12.23731834449151864806518400037, 12.80183041517583011647383130401
