L(s) = 1 | − 0.473i·2-s + 7.77·4-s + (−10.6 + 3.48i)5-s − 8.20i·7-s − 7.47i·8-s + (1.64 + 5.03i)10-s − 5.26·11-s + 77.0i·13-s − 3.88·14-s + 58.6·16-s − 88.9i·17-s + 91.7·19-s + (−82.6 + 27.0i)20-s + 2.49i·22-s + 154. i·23-s + ⋯ |
L(s) = 1 | − 0.167i·2-s + 0.971·4-s + (−0.950 + 0.311i)5-s − 0.443i·7-s − 0.330i·8-s + (0.0521 + 0.159i)10-s − 0.144·11-s + 1.64i·13-s − 0.0742·14-s + 0.916·16-s − 1.26i·17-s + 1.10·19-s + (−0.923 + 0.302i)20-s + 0.0241i·22-s + 1.40i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.053986354\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.053986354\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (10.6 - 3.48i)T \) |
good | 2 | \( 1 + 0.473iT - 8T^{2} \) |
| 7 | \( 1 + 8.20iT - 343T^{2} \) |
| 11 | \( 1 + 5.26T + 1.33e3T^{2} \) |
| 13 | \( 1 - 77.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 88.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 91.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 154. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 168.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 72.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 154. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 7.95T + 6.89e4T^{2} \) |
| 43 | \( 1 + 23.2iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 301. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 344. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 251.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 272.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 835. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 351.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 522. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 700.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 241. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 389. iT - 9.12e5T^{2} \) |
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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
−11.16214213240224488445914552147, −10.09822372354583757866557888041, −9.166960718336367041239039114662, −7.75325856660422570885664627205, −7.21420063353771336007388425586, −6.43973273704427732389547734834, −4.90091611485964781847661712012, −3.71156369714247161329768909417, −2.69204902782797785476753285465, −1.12023455635093209930695127748,
0.828897949149803896013174363248, 2.57355310935018226107201399724, 3.60111187189801712071253013088, 5.09770090519289561175703810635, 6.02243979005083481064955965392, 7.13616647195230287108638288182, 8.080834861520871573899606550287, 8.534391717658377293923877711077, 10.19770912385192263577053901491, 10.74322645379831194242323795430