L(s) = 1 | + (−1.16 − 3.82i)2-s + (−13.2 + 8.92i)4-s − 39.0·5-s − 12.2i·7-s + (49.6 + 40.4i)8-s + (45.5 + 149. i)10-s + 111. i·11-s + 208.·13-s + (−46.6 + 14.2i)14-s + (96.6 − 237. i)16-s + 93.3·17-s − 26.8i·19-s + (518. − 348. i)20-s + (424. − 129. i)22-s + 874. i·23-s + ⋯ |
L(s) = 1 | + (−0.291 − 0.956i)2-s + (−0.829 + 0.557i)4-s − 1.56·5-s − 0.249i·7-s + (0.775 + 0.631i)8-s + (0.455 + 1.49i)10-s + 0.917i·11-s + 1.23·13-s + (−0.238 + 0.0726i)14-s + (0.377 − 0.925i)16-s + 0.323·17-s − 0.0744i·19-s + (1.29 − 0.871i)20-s + (0.877 − 0.267i)22-s + 1.65i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 + 0.557i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.829 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5950543510\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5950543510\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.16 + 3.82i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 39.0T + 625T^{2} \) |
| 7 | \( 1 + 12.2iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 111. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 208.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 93.3T + 8.35e4T^{2} \) |
| 19 | \( 1 + 26.8iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 874. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.30e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 685. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.76e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 78.0T + 2.82e6T^{2} \) |
| 43 | \( 1 + 1.62e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.30e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.31e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 5.56e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.18e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 246. iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 4.60e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.56e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 5.18e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.87e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.16e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 5.73e3T + 8.85e7T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.86846724409766644689721759457, −9.755681680724063660787628415290, −8.768553426185071825496731658559, −7.82988148155087028023314516255, −7.18669956585527689689576772536, −5.26983275649394613466456284443, −3.91496662072985611375455085990, −3.55948704312845668881271806081, −1.70648808267101804980002262194, −0.27717247160335908242355890941,
0.923432797766425778454994357770, 3.42206849081103781550917391447, 4.30611477837558441478750673746, 5.61440430358903961577511935025, 6.62579015282706520480074541272, 7.67168642921440489683274086347, 8.443660588693679620645356308695, 8.965092590468941751034039677178, 10.54148547505496478908374109911, 11.18789394623724599437896253620