L(s) = 1 | + (0.739 − 2.27i)2-s + (−1.29 − 1.15i)3-s + (−3.01 − 2.19i)4-s + (−1.17 + 0.382i)5-s + (−3.58 + 2.08i)6-s + (−1.66 + 2.28i)7-s + (−3.35 + 2.43i)8-s + (0.334 + 2.98i)9-s + 2.96i·10-s + (1.36 + 6.31i)12-s + (−3.18 − 1.03i)13-s + (3.98 + 5.47i)14-s + (1.96 + 0.866i)15-s + (0.761 + 2.34i)16-s + (−1.28 − 3.94i)17-s + (7.03 + 1.44i)18-s + ⋯ |
L(s) = 1 | + (0.523 − 1.61i)2-s + (−0.745 − 0.666i)3-s + (−1.50 − 1.09i)4-s + (−0.527 + 0.171i)5-s + (−1.46 + 0.851i)6-s + (−0.628 + 0.864i)7-s + (−1.18 + 0.861i)8-s + (0.111 + 0.993i)9-s + 0.938i·10-s + (0.394 + 1.82i)12-s + (−0.882 − 0.286i)13-s + (1.06 + 1.46i)14-s + (0.507 + 0.223i)15-s + (0.190 + 0.585i)16-s + (−0.310 − 0.956i)17-s + (1.65 + 0.340i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.255248 + 0.207260i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.255248 + 0.207260i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.29 + 1.15i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.739 + 2.27i)T + (-1.61 - 1.17i)T^{2} \) |
| 5 | \( 1 + (1.17 - 0.382i)T + (4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (1.66 - 2.28i)T + (-2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (3.18 + 1.03i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.28 + 3.94i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (1.43 + 1.98i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (4.39 + 3.19i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.0606 + 0.186i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-8.46 - 6.15i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (5.81 - 4.22i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 4.24iT - 43T^{2} \) |
| 47 | \( 1 + (1.99 + 2.73i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (6.75 + 2.19i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-7.42 + 10.2i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (4.29 - 1.39i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + (-7.93 + 2.57i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.76 + 6.55i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (6.98 + 2.27i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (4.58 + 14.1i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 9.58iT - 89T^{2} \) |
| 97 | \( 1 + (-0.700 + 2.15i)T + (-78.4 - 57.0i)T^{2} \) |
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
−11.24204762508879064600046359226, −10.03700101431072928022728577178, −9.341409415149952559170055047974, −7.87908896062849654198380935863, −6.70336957504610907706998260369, −5.46779192104830113630451351686, −4.58290835062762405456212130507, −3.08978968709817227873903126978, −2.13058913598334370419908125811, −0.19875451690727844325977820831,
3.83643213492271848624858487040, 4.32217614251007445009417242804, 5.47462551982710000456680054852, 6.39580449771495129907463723125, 7.15880152521786831396674536195, 8.096456483177824221180482844338, 9.263265944001999724374920052401, 10.20948076104832972628011066207, 11.22461647047543639014016319050, 12.45290082364777998058876104504