L(s) = 1 | + (1.92 − 3.33i)2-s + (−3.41 − 5.91i)4-s + (2.5 + 4.33i)5-s + (13.1 − 22.8i)7-s + 4.52·8-s + 19.2·10-s + (2.53 − 4.39i)11-s + (41.3 + 71.6i)13-s + (−50.6 − 87.7i)14-s + (36.0 − 62.3i)16-s + 52.5·17-s − 29.8·19-s + (17.0 − 29.5i)20-s + (−9.77 − 16.9i)22-s + (49.1 + 85.1i)23-s + ⋯ |
L(s) = 1 | + (0.680 − 1.17i)2-s + (−0.426 − 0.738i)4-s + (0.223 + 0.387i)5-s + (0.710 − 1.23i)7-s + 0.199·8-s + 0.608·10-s + (0.0696 − 0.120i)11-s + (0.882 + 1.52i)13-s + (−0.967 − 1.67i)14-s + (0.562 − 0.974i)16-s + 0.750·17-s − 0.360·19-s + (0.190 − 0.330i)20-s + (−0.0947 − 0.164i)22-s + (0.445 + 0.771i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.486584679\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.486584679\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.5 - 4.33i)T \) |
good | 2 | \( 1 + (-1.92 + 3.33i)T + (-4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (-13.1 + 22.8i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-2.53 + 4.39i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-41.3 - 71.6i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 52.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 29.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-49.1 - 85.1i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-83.9 + 145. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (95.3 + 165. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 365.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (55.8 + 96.7i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-201. + 349. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-116. + 201. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 410.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-76.0 - 131. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (266. - 460. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (306. + 531. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 413.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 114.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (39.5 - 68.5i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (713. - 1.23e3i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 450.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (718. - 1.24e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−10.76187937837398580780816987411, −10.16097585013009424376298878599, −8.989057579863817091953366468073, −7.66649493390172475492038250987, −6.84387978895466757819976549888, −5.43317574337670770859854661888, −4.15396517368751670373484120546, −3.70208421303497943931356888372, −2.11844280633748984754206853028, −1.13061778395147925968531348964,
1.43178698246530459002536021264, 3.14814590930375470168181931129, 4.71387589198935980207559107384, 5.45589062313432219571450505213, 6.02859058872587507939314634020, 7.24299134190912976415145657165, 8.392193332571037973416730256380, 8.660853117388097551065833302907, 10.23705555441596841602969012239, 11.07382501863723961579322836036