L(s) = 1 | + (−0.720 + 2.68i)2-s − i·3-s + (−4.97 − 2.87i)4-s + (−0.470 − 1.75i)5-s + (2.68 + 0.720i)6-s + (1.29 + 2.30i)7-s + (7.38 − 7.38i)8-s − 9-s + 5.05·10-s + (3.13 − 3.13i)11-s + (−2.87 + 4.97i)12-s + (2.12 + 2.91i)13-s + (−7.13 + 1.81i)14-s + (−1.75 + 0.470i)15-s + (8.78 + 15.2i)16-s + (0.0321 − 0.0557i)17-s + ⋯ |
L(s) = 1 | + (−0.509 + 1.90i)2-s − 0.577i·3-s + (−2.48 − 1.43i)4-s + (−0.210 − 0.784i)5-s + (1.09 + 0.294i)6-s + (0.489 + 0.872i)7-s + (2.61 − 2.61i)8-s − 0.333·9-s + 1.59·10-s + (0.945 − 0.945i)11-s + (−0.829 + 1.43i)12-s + (0.590 + 0.807i)13-s + (−1.90 + 0.486i)14-s + (−0.452 + 0.121i)15-s + (2.19 + 3.80i)16-s + (0.00780 − 0.0135i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.504 - 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.779391 + 0.447316i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.779391 + 0.447316i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 + (-1.29 - 2.30i)T \) |
| 13 | \( 1 + (-2.12 - 2.91i)T \) |
good | 2 | \( 1 + (0.720 - 2.68i)T + (-1.73 - i)T^{2} \) |
| 5 | \( 1 + (0.470 + 1.75i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-3.13 + 3.13i)T - 11iT^{2} \) |
| 17 | \( 1 + (-0.0321 + 0.0557i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.15 + 2.15i)T - 19iT^{2} \) |
| 23 | \( 1 + (-2.78 + 1.60i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.41 + 4.18i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (5.02 + 1.34i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-4.21 - 1.12i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (2.12 + 7.93i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (3.22 - 1.86i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.06 + 1.08i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.30 - 5.71i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.49 - 1.20i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 - 2.19iT - 61T^{2} \) |
| 67 | \( 1 + (-4.45 - 4.45i)T + 67iT^{2} \) |
| 71 | \( 1 + (1.98 - 7.40i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-1.61 + 6.03i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-0.639 + 1.10i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (8.90 - 8.90i)T - 83iT^{2} \) |
| 89 | \( 1 + (-0.376 + 1.40i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (8.79 + 2.35i)T + (84.0 + 48.5i)T^{2} \) |
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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
−12.24937471907196571303160732737, −11.12823016682140512621405967912, −9.376047317628816483756316711803, −8.766505176097071719872919306322, −8.323119111350253660259080000995, −7.12636240503533667048295957302, −6.20340705299000332385522245629, −5.40076829021570012251087545250, −4.26040568400638939405159927442, −1.06839343885753982675970448751,
1.40132255821710512781968716882, 3.15235001219902838085569303750, 3.89498006895850334325139496491, 4.97325640521922652019577073490, 7.21271210527102159918258757654, 8.256398206457793576418350130915, 9.356634987345574310571014412470, 10.12723401445139496368399301989, 10.85164224007341529603917597247, 11.35153969434730669383727894394