| 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
−10.20107510846580342265967039527, −9.435163801932549986119119461707, −8.776938474363162479194463184821, −7.60928013193313848901268143602, −6.46480019596757851100518070979, −5.97933274136923703753365860652, −5.18270117459005376117503905099, −4.07308411067498043023807743753, −2.96032162448834672403941053183, −1.98745199559896672521757805574,
1.15839549610016741628158784790, 2.13998064083922070159792548712, 3.22535818974214353377437641293, 4.52339723564125625126360351764, 5.65083214458202449315456343599, 6.05800279797636648847008444175, 7.04166618974560922349378968030, 8.115522152765642073621825908810, 9.216677124700757906869446343947, 9.780927403214123357462907790029
