| L(s) = 1 | + (−2.25 − 1.05i)2-s + (0.997 − 1.41i)3-s + (2.69 + 3.21i)4-s + (2.22 − 0.262i)5-s + (−3.74 + 2.14i)6-s + (−0.697 − 0.186i)7-s + (−1.41 − 5.27i)8-s + (−1.00 − 2.82i)9-s + (−5.28 − 1.74i)10-s + (5.45 + 3.15i)11-s + (7.24 − 0.610i)12-s + (0.863 − 1.23i)13-s + (1.37 + 1.15i)14-s + (1.84 − 3.40i)15-s + (−0.905 + 5.13i)16-s + (1.79 + 0.835i)17-s + ⋯ |
| L(s) = 1 | + (−1.59 − 0.743i)2-s + (0.576 − 0.817i)3-s + (1.34 + 1.60i)4-s + (0.993 − 0.117i)5-s + (−1.52 + 0.875i)6-s + (−0.263 − 0.0706i)7-s + (−0.500 − 1.86i)8-s + (−0.336 − 0.941i)9-s + (−1.67 − 0.551i)10-s + (1.64 + 0.949i)11-s + (2.09 − 0.176i)12-s + (0.239 − 0.342i)13-s + (0.368 + 0.308i)14-s + (0.476 − 0.879i)15-s + (−0.226 + 1.28i)16-s + (0.434 + 0.202i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0267 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0267 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.628679 - 0.645725i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.628679 - 0.645725i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.997 + 1.41i)T \) |
| 5 | \( 1 + (-2.22 + 0.262i)T \) |
| 19 | \( 1 + (-0.484 - 4.33i)T \) |
| good | 2 | \( 1 + (2.25 + 1.05i)T + (1.28 + 1.53i)T^{2} \) |
| 7 | \( 1 + (0.697 + 0.186i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-5.45 - 3.15i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.863 + 1.23i)T + (-4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.79 - 0.835i)T + (10.9 + 13.0i)T^{2} \) |
| 23 | \( 1 + (-0.423 + 4.84i)T + (-22.6 - 3.99i)T^{2} \) |
| 29 | \( 1 + (8.85 + 3.22i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (1.83 + 3.17i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.58 - 2.58i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.19 - 0.210i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.171 - 1.95i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (2.08 + 4.47i)T + (-30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (0.668 - 7.64i)T + (-52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (-4.93 + 1.79i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (4.08 - 3.42i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (0.0148 - 0.00692i)T + (43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (-3.88 + 4.63i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (10.8 - 7.59i)T + (24.9 - 68.5i)T^{2} \) |
| 79 | \( 1 + (-11.0 - 1.94i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.658 - 0.176i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-1.53 - 8.67i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-3.02 + 6.48i)T + (-62.3 - 74.3i)T^{2} \) |
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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
−11.56273217715381005519368976778, −10.26786569132513283324770460689, −9.547968802063078436249173705894, −9.005011982547046997846697474810, −8.001040414242426395842443119925, −7.00686970322702274810459270319, −6.11561642548778523608118674108, −3.63227287024579040303870418028, −2.16188902763376734361918477194, −1.30840016810617702706124964577,
1.63225085847790364391841163333, 3.43680530527617922873311116549, 5.41589156705825808387619845709, 6.37695926220822472276987598401, 7.35140140107985682739988799771, 8.709536731914030721904091097066, 9.241537068220842867140873735857, 9.607463580174080490988077159433, 10.78231541769762762572462526839, 11.40311761338515272491938519343
