L(s) = 1 | + (−1.17 − 0.785i)2-s + (−1.01 − 1.40i)3-s + (0.764 + 1.84i)4-s + (2.85 + 0.376i)5-s + (0.0905 + 2.44i)6-s + (−0.838 + 3.13i)7-s + (0.553 − 2.77i)8-s + (−0.939 + 2.84i)9-s + (−3.06 − 2.68i)10-s + (−3.69 + 4.81i)11-s + (1.81 − 2.94i)12-s + (−3.49 + 2.68i)13-s + (3.44 − 3.02i)14-s + (−2.37 − 4.39i)15-s + (−2.83 + 2.82i)16-s + 4.67·17-s + ⋯ |
L(s) = 1 | + (−0.831 − 0.555i)2-s + (−0.586 − 0.810i)3-s + (0.382 + 0.924i)4-s + (1.27 + 0.168i)5-s + (0.0369 + 0.999i)6-s + (−0.317 + 1.18i)7-s + (0.195 − 0.980i)8-s + (−0.313 + 0.949i)9-s + (−0.968 − 0.849i)10-s + (−1.11 + 1.45i)11-s + (0.524 − 0.851i)12-s + (−0.970 + 0.744i)13-s + (0.921 − 0.807i)14-s + (−0.612 − 1.13i)15-s + (−0.707 + 0.706i)16-s + 1.13·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.649421 + 0.215467i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.649421 + 0.215467i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.17 + 0.785i)T \) |
| 3 | \( 1 + (1.01 + 1.40i)T \) |
good | 5 | \( 1 + (-2.85 - 0.376i)T + (4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (0.838 - 3.13i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (3.69 - 4.81i)T + (-2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (3.49 - 2.68i)T + (3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 - 4.67T + 17T^{2} \) |
| 19 | \( 1 + (1.17 - 2.83i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-3.57 + 0.957i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (0.568 + 4.31i)T + (-28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (3.52 + 2.03i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.98 + 0.820i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-0.0956 - 0.357i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-0.627 - 0.481i)T + (11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (-2.98 + 1.72i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.59 - 2.73i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-0.174 + 1.32i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (2.52 - 0.331i)T + (58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (-8.04 + 6.17i)T + (17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (-5.75 + 5.75i)T - 71iT^{2} \) |
| 73 | \( 1 + (-11.3 - 11.3i)T + 73iT^{2} \) |
| 79 | \( 1 + (-7.01 - 12.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.31 + 17.6i)T + (-80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (-2.22 - 2.22i)T + 89iT^{2} \) |
| 97 | \( 1 + (-0.970 - 1.68i)T + (-48.5 + 84.0i)T^{2} \) |
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
−12.17397511160613693536499950387, −10.86791978940493404695766506248, −9.847302482268427260345743075800, −9.476533904911375038032125038069, −8.015761841017350197482788638845, −7.14388101632812149469492555496, −6.07623964830356555891312912019, −5.06887096277029933373717546659, −2.48679755764171339769438507448, −1.99220987431463163153813125628,
0.69660154553572047428425693000, 3.05893605330352143802147444969, 5.15650981968217950173977646162, 5.59266555466031004213371553355, 6.73607425652437579666956072247, 7.88404252608594341429855585716, 9.138420907755777101284888360650, 9.911453912942338380356566173648, 10.54490017556726338968673191438, 11.04577271959173000709721263294