| L(s) = 1 | + (−0.959 + 0.281i)3-s + (−1.03 − 0.663i)5-s + (−0.0639 − 0.444i)7-s + (0.841 − 0.540i)9-s + (−0.763 + 1.67i)11-s + (0.0274 − 0.190i)13-s + (1.17 + 0.345i)15-s + (−4.88 − 5.63i)17-s + (−1.51 + 1.74i)19-s + (0.186 + 0.408i)21-s + (−4.79 + 0.0735i)23-s + (−1.45 − 3.17i)25-s + (−0.654 + 0.755i)27-s + (−1.04 − 1.20i)29-s + (−6.45 − 1.89i)31-s + ⋯ |
| L(s) = 1 | + (−0.553 + 0.162i)3-s + (−0.461 − 0.296i)5-s + (−0.0241 − 0.168i)7-s + (0.280 − 0.180i)9-s + (−0.230 + 0.503i)11-s + (0.00761 − 0.0529i)13-s + (0.303 + 0.0892i)15-s + (−1.18 − 1.36i)17-s + (−0.347 + 0.401i)19-s + (0.0407 + 0.0892i)21-s + (−0.999 + 0.0153i)23-s + (−0.290 − 0.635i)25-s + (−0.126 + 0.145i)27-s + (−0.193 − 0.223i)29-s + (−1.15 − 0.340i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0888633 - 0.308180i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0888633 - 0.308180i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (4.79 - 0.0735i)T \) |
| good | 5 | \( 1 + (1.03 + 0.663i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (0.0639 + 0.444i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (0.763 - 1.67i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.0274 + 0.190i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (4.88 + 5.63i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (1.51 - 1.74i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (1.04 + 1.20i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (6.45 + 1.89i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-0.540 + 0.347i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (4.22 + 2.71i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-6.95 + 2.04i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 8.78T + 47T^{2} \) |
| 53 | \( 1 + (1.18 + 8.24i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (1.37 - 9.59i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (0.818 + 0.240i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-1.62 - 3.55i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-0.316 - 0.692i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (6.35 - 7.33i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (0.576 - 4.00i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-6.08 + 3.91i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-2.35 + 0.691i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-9.01 - 5.79i)T + (40.2 + 88.2i)T^{2} \) |
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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
−10.45120108921112338887931565681, −9.646145753452973221745008478783, −8.693627711666927012133561079612, −7.64096470841792007275138090770, −6.84131144267063528696715391656, −5.72675911782525199668076789396, −4.68605352984305835303609383663, −3.89965491611293979837500827454, −2.20600796286830652386254105906, −0.18706357071958417586510798978,
1.92442909030124633946298372568, 3.49723969729753833454551034584, 4.53910936056135995422199256139, 5.78726482573834936441407922210, 6.51020565026464397641035373323, 7.54825258382646627193581337221, 8.424746129458966869276835911659, 9.361444729545345074654157138833, 10.65570707552658880390753435389, 10.99277305293079075009780544568
