L(s) = 1 | + (1.43 − 2.43i)2-s + (5.17 − 0.459i)3-s + (−3.85 − 7.00i)4-s + (−8.70 − 7.01i)5-s + (6.32 − 13.2i)6-s + 7.71·7-s + (−22.6 − 0.684i)8-s + (26.5 − 4.75i)9-s + (−29.6 + 11.0i)10-s + 38.0·11-s + (−23.1 − 34.4i)12-s + 63.3i·13-s + (11.1 − 18.7i)14-s + (−48.2 − 32.3i)15-s + (−34.2 + 54.0i)16-s − 10.2·17-s + ⋯ |
L(s) = 1 | + (0.508 − 0.860i)2-s + (0.996 − 0.0883i)3-s + (−0.482 − 0.875i)4-s + (−0.778 − 0.627i)5-s + (0.430 − 0.902i)6-s + 0.416·7-s + (−0.999 − 0.0302i)8-s + (0.984 − 0.176i)9-s + (−0.936 + 0.350i)10-s + 1.04·11-s + (−0.557 − 0.829i)12-s + 1.35i·13-s + (0.211 − 0.358i)14-s + (−0.830 − 0.556i)15-s + (−0.534 + 0.845i)16-s − 0.146·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0867 + 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0867 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.44984 - 1.58156i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44984 - 1.58156i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.43 + 2.43i)T \) |
| 3 | \( 1 + (-5.17 + 0.459i)T \) |
| 5 | \( 1 + (8.70 + 7.01i)T \) |
good | 7 | \( 1 - 7.71T + 343T^{2} \) |
| 11 | \( 1 - 38.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 63.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 10.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 99.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 133. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 197. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 13.9iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 272. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 166. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 273.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 69.1iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 300.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 618.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 439.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 23.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + 827.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 152. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 421. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 709. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.43e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.08e3iT - 9.12e5T^{2} \) |
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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
−14.17329969236375731411241081779, −13.20019219914484003558657586023, −12.03663905860963546293931452579, −11.20874774010723949749325645907, −9.356328647392835804823336703193, −8.790631580980733744424966128373, −7.03945396530230334948168047843, −4.70326113411420348772653976309, −3.64373878541747859419306630885, −1.59317436900991581330051552794,
3.18480818316974661144971948821, 4.39059525601661110827676655019, 6.45103297466473964286113320320, 7.80646398908499777391486504335, 8.426111425723259578066150728203, 10.08360027473381779142482218679, 11.78218597877962781785975836567, 12.92613200603576803615663146517, 14.22492849579717082102814058670, 14.81352489649298498530955770869