| L(s) = 1 | + (0.841 − 0.540i)2-s + (0.142 − 0.989i)3-s + (0.415 − 0.909i)4-s + (3.26 − 0.958i)5-s + (−0.415 − 0.909i)6-s + (−0.654 − 0.755i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (2.22 − 2.57i)10-s + (0.242 + 0.155i)11-s + (−0.841 − 0.540i)12-s + (1.17 − 1.35i)13-s + (−0.959 − 0.281i)14-s + (−0.484 − 3.36i)15-s + (−0.654 − 0.755i)16-s + (0.365 + 0.801i)17-s + ⋯ |
| L(s) = 1 | + (0.594 − 0.382i)2-s + (0.0821 − 0.571i)3-s + (0.207 − 0.454i)4-s + (1.46 − 0.428i)5-s + (−0.169 − 0.371i)6-s + (−0.247 − 0.285i)7-s + (−0.0503 − 0.349i)8-s + (−0.319 − 0.0939i)9-s + (0.704 − 0.813i)10-s + (0.0730 + 0.0469i)11-s + (−0.242 − 0.156i)12-s + (0.325 − 0.375i)13-s + (−0.256 − 0.0752i)14-s + (−0.125 − 0.869i)15-s + (−0.163 − 0.188i)16-s + (0.0887 + 0.194i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.80343 - 2.16300i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80343 - 2.16300i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 + 0.540i)T \) |
| 3 | \( 1 + (-0.142 + 0.989i)T \) |
| 7 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (3.13 - 3.63i)T \) |
| good | 5 | \( 1 + (-3.26 + 0.958i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (-0.242 - 0.155i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.17 + 1.35i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.365 - 0.801i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.787 + 1.72i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (1.76 + 3.87i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.596 - 4.14i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-2.02 - 0.595i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-3.99 + 1.17i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (0.801 - 5.57i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 2.61T + 47T^{2} \) |
| 53 | \( 1 + (0.982 + 1.13i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (6.90 - 7.97i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.02 + 7.15i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (1.45 - 0.936i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-1.64 + 1.05i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (3.39 - 7.44i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (0.585 - 0.676i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-3.12 - 0.917i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (0.0869 - 0.605i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-2.39 + 0.702i)T + (81.6 - 52.4i)T^{2} \) |
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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
−9.780927403214123357462907790029, −9.216677124700757906869446343947, −8.115522152765642073621825908810, −7.04166618974560922349378968030, −6.05800279797636648847008444175, −5.65083214458202449315456343599, −4.52339723564125625126360351764, −3.22535818974214353377437641293, −2.13998064083922070159792548712, −1.15839549610016741628158784790,
1.98745199559896672521757805574, 2.96032162448834672403941053183, 4.07308411067498043023807743753, 5.18270117459005376117503905099, 5.97933274136923703753365860652, 6.46480019596757851100518070979, 7.60928013193313848901268143602, 8.776938474363162479194463184821, 9.435163801932549986119119461707, 10.20107510846580342265967039527
