L(s) = 1 | + (−2.82 − 0.218i)2-s + (−2.56 − 4.51i)3-s + (7.90 + 1.23i)4-s + (2.35 + 6.47i)5-s + (6.24 + 13.3i)6-s + (−6.19 + 1.09i)7-s + (−22.0 − 5.19i)8-s + (−13.8 + 23.1i)9-s + (−5.23 − 18.7i)10-s + (−20.4 − 7.42i)11-s + (−14.7 − 38.8i)12-s + (59.5 + 49.9i)13-s + (17.7 − 1.73i)14-s + (23.2 − 27.2i)15-s + (60.9 + 19.4i)16-s + (34.1 + 19.6i)17-s + ⋯ |
L(s) = 1 | + (−0.997 − 0.0771i)2-s + (−0.493 − 0.869i)3-s + (0.988 + 0.153i)4-s + (0.210 + 0.578i)5-s + (0.425 + 0.905i)6-s + (−0.334 + 0.0590i)7-s + (−0.973 − 0.229i)8-s + (−0.512 + 0.858i)9-s + (−0.165 − 0.593i)10-s + (−0.559 − 0.203i)11-s + (−0.354 − 0.935i)12-s + (1.27 + 1.06i)13-s + (0.338 − 0.0330i)14-s + (0.399 − 0.469i)15-s + (0.952 + 0.303i)16-s + (0.486 + 0.280i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.394i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.786139 + 0.161651i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.786139 + 0.161651i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.82 + 0.218i)T \) |
| 3 | \( 1 + (2.56 + 4.51i)T \) |
good | 5 | \( 1 + (-2.35 - 6.47i)T + (-95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (6.19 - 1.09i)T + (322. - 117. i)T^{2} \) |
| 11 | \( 1 + (20.4 + 7.42i)T + (1.01e3 + 855. i)T^{2} \) |
| 13 | \( 1 + (-59.5 - 49.9i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-34.1 - 19.6i)T + (2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-102. + 59.4i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (29.8 - 169. i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-13.5 - 16.1i)T + (-4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-123. - 21.7i)T + (2.79e4 + 1.01e4i)T^{2} \) |
| 37 | \( 1 + (100. - 173. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-154. + 184. i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (20.3 - 55.7i)T + (-6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-45.4 - 257. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + 26.8iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-104. + 37.8i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-105. - 597. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (506. - 603. i)T + (-5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (76.4 - 132. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (273. + 472. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-103. - 123. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (889. - 746. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-1.20e3 + 697. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.25e3 - 457. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−13.23008177129490898198447078124, −11.86802863122121136464732094557, −11.22498743190534129040108323159, −10.18691208213082763251402559209, −8.907699320437728468003975390140, −7.68835751802056164569857190196, −6.72309994830468907699630903201, −5.78421413075877792236429290775, −2.98569531199364680800807409872, −1.30991853260479264456107591518,
0.74788959590032162780477521999, 3.25745233477179283253796812936, 5.25840907249757994343375636924, 6.25222936390735685630446264400, 7.943042152431423214924170883531, 8.967652927991033934228063266177, 10.03420418726656267725450719131, 10.64925514455822015277108945698, 11.85000289930723061856562401985, 12.88678133455330685546097833205