L(s) = 1 | + (1.00 − 1.73i)2-s + (−1.98 − 3.47i)4-s + (−0.608 + 3.44i)5-s + (5.71 + 6.81i)7-s + (−7.99 − 0.0425i)8-s + (5.35 + 4.51i)10-s + (5.38 − 0.949i)11-s + (21.3 − 7.76i)13-s + (17.5 − 3.05i)14-s + (−8.09 + 13.7i)16-s + (8.12 − 14.0i)17-s + (19.5 − 11.2i)19-s + (13.1 − 4.74i)20-s + (3.75 − 10.2i)22-s + (−22.1 + 26.4i)23-s + ⋯ |
L(s) = 1 | + (0.501 − 0.865i)2-s + (−0.496 − 0.867i)4-s + (−0.121 + 0.689i)5-s + (0.816 + 0.973i)7-s + (−0.999 − 0.00532i)8-s + (0.535 + 0.451i)10-s + (0.489 − 0.0862i)11-s + (1.64 − 0.597i)13-s + (1.25 − 0.218i)14-s + (−0.506 + 0.862i)16-s + (0.478 − 0.827i)17-s + (1.02 − 0.593i)19-s + (0.658 − 0.237i)20-s + (0.170 − 0.466i)22-s + (−0.963 + 1.14i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.11223 - 0.867464i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.11223 - 0.867464i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.00 + 1.73i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.608 - 3.44i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-5.71 - 6.81i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-5.38 + 0.949i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-21.3 + 7.76i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-8.12 + 14.0i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-19.5 + 11.2i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (22.1 - 26.4i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (24.1 + 8.78i)T + (644. + 540. i)T^{2} \) |
| 31 | \( 1 + (14.3 - 17.1i)T + (-166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-7.88 + 13.6i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-49.6 + 18.0i)T + (1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (9.61 - 1.69i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-14.9 - 17.8i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 4.09T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-28.0 - 4.93i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (31.1 - 26.1i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-30.9 - 84.9i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (87.8 + 50.7i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (16.0 + 27.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-8.65 + 23.7i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (16.3 - 45.0i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (45.0 + 78.0i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (25.7 + 146. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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
−11.42527236162146538022519070525, −10.71369067633856745677019998271, −9.483016967412746024967419264012, −8.732565584999136572940741410402, −7.46613876871099874745094960046, −5.95581747051288226765512776149, −5.33451989472520059958980918873, −3.81270351769672112101810347754, −2.83769248859334137364046925467, −1.35353605804153472786046468611,
1.25716769532721359041916976874, 3.78644846604384312484822248219, 4.32878176064128170746258082521, 5.61870458282541463449108460905, 6.57458590421313596437252119156, 7.78967245563741958627756347371, 8.363583704974254619839942831470, 9.340945200272408221793779376456, 10.71732021015984220728329595661, 11.67427642352361966728892958737