| L(s) = 1 | + (0.655 + 1.23i)2-s + (−0.725 − 0.687i)3-s + (0.0222 − 0.0327i)4-s + (2.32 + 0.512i)5-s + (0.374 − 1.34i)6-s + (−1.36 − 1.03i)7-s + (2.83 + 0.308i)8-s + (0.0541 + 0.998i)9-s + (0.892 + 3.21i)10-s + (0.846 − 2.12i)11-s + (−0.0386 + 0.00851i)12-s + (−0.224 + 4.13i)13-s + (0.387 − 2.36i)14-s + (−1.33 − 1.97i)15-s + (1.45 + 3.64i)16-s + (−2.30 + 1.75i)17-s + ⋯ |
| L(s) = 1 | + (0.463 + 0.874i)2-s + (−0.419 − 0.397i)3-s + (0.0111 − 0.0163i)4-s + (1.04 + 0.229i)5-s + (0.152 − 0.550i)6-s + (−0.515 − 0.391i)7-s + (1.00 + 0.109i)8-s + (0.0180 + 0.332i)9-s + (0.282 + 1.01i)10-s + (0.255 − 0.640i)11-s + (−0.0111 + 0.00245i)12-s + (−0.0622 + 1.14i)13-s + (0.103 − 0.632i)14-s + (−0.345 − 0.509i)15-s + (0.362 + 0.910i)16-s + (−0.559 + 0.425i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.46553 + 0.428917i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46553 + 0.428917i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.725 + 0.687i)T \) |
| 59 | \( 1 + (-5.56 + 5.29i)T \) |
| good | 2 | \( 1 + (-0.655 - 1.23i)T + (-1.12 + 1.65i)T^{2} \) |
| 5 | \( 1 + (-2.32 - 0.512i)T + (4.53 + 2.09i)T^{2} \) |
| 7 | \( 1 + (1.36 + 1.03i)T + (1.87 + 6.74i)T^{2} \) |
| 11 | \( 1 + (-0.846 + 2.12i)T + (-7.98 - 7.56i)T^{2} \) |
| 13 | \( 1 + (0.224 - 4.13i)T + (-12.9 - 1.40i)T^{2} \) |
| 17 | \( 1 + (2.30 - 1.75i)T + (4.54 - 16.3i)T^{2} \) |
| 19 | \( 1 + (1.04 + 0.352i)T + (15.1 + 11.4i)T^{2} \) |
| 23 | \( 1 + (4.30 + 2.58i)T + (10.7 + 20.3i)T^{2} \) |
| 29 | \( 1 + (-2.49 + 4.70i)T + (-16.2 - 24.0i)T^{2} \) |
| 31 | \( 1 + (2.74 - 0.923i)T + (24.6 - 18.7i)T^{2} \) |
| 37 | \( 1 + (9.84 - 1.07i)T + (36.1 - 7.95i)T^{2} \) |
| 41 | \( 1 + (3.19 - 1.92i)T + (19.2 - 36.2i)T^{2} \) |
| 43 | \( 1 + (-0.835 - 2.09i)T + (-31.2 + 29.5i)T^{2} \) |
| 47 | \( 1 + (-7.54 + 1.66i)T + (42.6 - 19.7i)T^{2} \) |
| 53 | \( 1 + (1.24 - 4.46i)T + (-45.4 - 27.3i)T^{2} \) |
| 61 | \( 1 + (-5.85 - 11.0i)T + (-34.2 + 50.4i)T^{2} \) |
| 67 | \( 1 + (8.79 + 0.955i)T + (65.4 + 14.4i)T^{2} \) |
| 71 | \( 1 + (8.56 - 1.88i)T + (64.4 - 29.8i)T^{2} \) |
| 73 | \( 1 + (-0.561 + 3.42i)T + (-69.1 - 23.3i)T^{2} \) |
| 79 | \( 1 + (9.43 - 8.94i)T + (4.27 - 78.8i)T^{2} \) |
| 83 | \( 1 + (-11.2 - 13.2i)T + (-13.4 + 81.9i)T^{2} \) |
| 89 | \( 1 + (-0.366 + 0.691i)T + (-49.9 - 73.6i)T^{2} \) |
| 97 | \( 1 + (0.438 + 2.67i)T + (-91.9 + 30.9i)T^{2} \) |
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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
−13.23779602268424429180297229122, −11.88656427183081338335292685037, −10.71720928666155914673909901901, −9.941327975867156886621089857471, −8.560630109478562617339728289375, −7.03583008282633766472343959746, −6.41081007767067724251643746868, −5.70427953131167883419969911015, −4.25193976871656730016622471547, −1.95126009859808909161875779550,
2.02283208035468353216074910425, 3.43873189172604863460756517168, 4.89532757285123338087024435144, 5.91488156754350747121445242016, 7.26849064603097834232295636399, 8.925098673499019803246097149792, 9.999866129583441086959226716073, 10.54318523212128475062973489822, 11.79400751840570998738781263466, 12.55018660778418223989118236542
