| L(s) = 1 | + (−0.445 + 1.94i)2-s + (3.18 − 3.99i)3-s + (−3.60 − 1.73i)4-s + (−8.89 + 11.1i)5-s + (6.37 + 7.99i)6-s + (−18.5 + 0.691i)7-s + (4.98 − 6.25i)8-s + (0.196 + 0.859i)9-s + (−17.7 − 22.3i)10-s + (−8.67 + 38.0i)11-s + (−18.4 + 8.86i)12-s + (−12.7 + 55.7i)13-s + (6.88 − 36.3i)14-s + (16.2 + 71.0i)15-s + (9.97 + 12.5i)16-s + (69.2 − 33.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.157 + 0.689i)2-s + (0.613 − 0.768i)3-s + (−0.450 − 0.216i)4-s + (−0.795 + 0.997i)5-s + (0.433 + 0.543i)6-s + (−0.999 + 0.0373i)7-s + (0.220 − 0.276i)8-s + (0.00726 + 0.0318i)9-s + (−0.562 − 0.705i)10-s + (−0.237 + 1.04i)11-s + (−0.443 + 0.213i)12-s + (−0.271 + 1.18i)13-s + (0.131 − 0.694i)14-s + (0.279 + 1.22i)15-s + (0.155 + 0.195i)16-s + (0.988 − 0.476i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.536i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.228696 + 0.785729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.228696 + 0.785729i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.445 - 1.94i)T \) |
| 7 | \( 1 + (18.5 - 0.691i)T \) |
| good | 3 | \( 1 + (-3.18 + 3.99i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (8.89 - 11.1i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (8.67 - 38.0i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (12.7 - 55.7i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (-69.2 + 33.3i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 + 52.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + (189. + 91.4i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (-176. + 85.1i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 + 291.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-219. + 105. i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (8.69 - 10.9i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (-292. - 366. i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (28.6 - 125. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-140. - 67.5i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (-61.2 - 76.8i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (350. - 168. i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 + 127.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-157. - 75.9i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (-141. - 619. i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 - 113.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (251. + 1.10e3i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-176. - 771. i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + 118.T + 9.12e5T^{2} \) |
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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
−14.21089239677228551155655852401, −12.90705051941495162833318984583, −12.03166790553520802046478235557, −10.43948809304921528899999944436, −9.381127635846380066470806387721, −7.87945293924230713519131029934, −7.24189365074513116644349128794, −6.35491074385554748459215039268, −4.18074373957838485858574084174, −2.47921355496781827786710917810,
0.45145916025298552581285515072, 3.22107841698431109519370766856, 3.99830770459949483140987536034, 5.66934138290639773682018815271, 7.940604704091861559064873908275, 8.716172597102488634881626892644, 9.771598367297464649944949644962, 10.61079882344396161432478305386, 12.17779371729692849388017807538, 12.70133852555271460602238991279
