| L(s) = 1 | + (−0.451 + 1.34i)2-s + (−0.520 − 2.95i)3-s + (−1.59 − 1.21i)4-s + (−4.82 − 1.92i)5-s + (4.19 + 0.636i)6-s + (−5.47 + 2.53i)7-s + (2.34 − 1.58i)8-s + (−8.45 + 3.07i)9-s + (4.75 − 5.59i)10-s + (9.66 + 2.68i)11-s + (−2.74 + 5.33i)12-s + (3.44 − 6.49i)13-s + (−0.923 − 8.48i)14-s + (−3.16 + 15.2i)15-s + (1.07 + 3.85i)16-s + (−4.82 + 10.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.225 + 0.670i)2-s + (−0.173 − 0.984i)3-s + (−0.398 − 0.302i)4-s + (−0.965 − 0.384i)5-s + (0.699 + 0.106i)6-s + (−0.782 + 0.362i)7-s + (0.292 − 0.198i)8-s + (−0.939 + 0.341i)9-s + (0.475 − 0.559i)10-s + (0.878 + 0.243i)11-s + (−0.228 + 0.444i)12-s + (0.264 − 0.499i)13-s + (−0.0659 − 0.606i)14-s + (−0.211 + 1.01i)15-s + (0.0668 + 0.240i)16-s + (−0.283 + 0.613i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.508290 + 0.425885i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.508290 + 0.425885i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.451 - 1.34i)T \) |
| 3 | \( 1 + (0.520 + 2.95i)T \) |
| 59 | \( 1 + (-4.12 - 58.8i)T \) |
| good | 5 | \( 1 + (4.82 + 1.92i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (5.47 - 2.53i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (-9.66 - 2.68i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (-3.44 + 6.49i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (4.82 - 10.4i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (11.5 - 2.53i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (-41.0 - 6.72i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (-13.4 - 39.8i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (-8.35 - 1.83i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (17.8 - 26.2i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (-69.5 + 11.4i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (-7.23 - 26.0i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (28.2 - 11.2i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (3.54 - 3.00i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (68.5 + 23.0i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (38.2 + 56.4i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (57.9 - 23.0i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (-60.2 + 6.55i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (50.7 - 30.5i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (134. - 7.27i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (-6.31 - 18.7i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (-96.6 - 10.5i)T + (9.18e3 + 2.02e3i)T^{2} \) |
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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
−11.58943606697924496610561824759, −10.64648533133093955377207770381, −9.140806497425524865605276738228, −8.573986483913572799827238926909, −7.59741048500885043422134771538, −6.72727361550220190408663847353, −5.97963704627499717093498967991, −4.63645390926989911033361148927, −3.18819800171061053558240551717, −1.14395448486098182332492854668,
0.40381959719245156144606730382, 2.91222480165135641103771368166, 3.83656839829674973956058278882, 4.56517931408260961516777460262, 6.23660710135509859257598428633, 7.26228936291465703112229701937, 8.662000428132704561160131357894, 9.296144467482166057335033180551, 10.21201867252860247351787695069, 11.24796027862196057108958513078
