L(s) = 1 | + (−0.719 − 2.73i)2-s + (−2.66 − 1.10i)3-s + (−6.96 + 3.93i)4-s + (−5.50 − 13.2i)5-s + (−1.09 + 8.07i)6-s + (−6.48 + 6.48i)7-s + (15.7 + 16.2i)8-s + (−13.2 − 13.2i)9-s + (−32.4 + 24.6i)10-s + (49.3 − 20.4i)11-s + (22.8 − 2.80i)12-s + (21.4 − 51.7i)13-s + (22.3 + 13.0i)14-s + 41.4i·15-s + (32.9 − 54.8i)16-s − 3.73i·17-s + ⋯ |
L(s) = 1 | + (−0.254 − 0.967i)2-s + (−0.512 − 0.212i)3-s + (−0.870 + 0.492i)4-s + (−0.492 − 1.18i)5-s + (−0.0748 + 0.549i)6-s + (−0.349 + 0.349i)7-s + (0.697 + 0.716i)8-s + (−0.489 − 0.489i)9-s + (−1.02 + 0.779i)10-s + (1.35 − 0.560i)11-s + (0.550 − 0.0675i)12-s + (0.457 − 1.10i)13-s + (0.427 + 0.249i)14-s + 0.714i·15-s + (0.515 − 0.857i)16-s − 0.0533i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.101406 - 0.659553i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.101406 - 0.659553i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.719 + 2.73i)T \) |
good | 3 | \( 1 + (2.66 + 1.10i)T + (19.0 + 19.0i)T^{2} \) |
| 5 | \( 1 + (5.50 + 13.2i)T + (-88.3 + 88.3i)T^{2} \) |
| 7 | \( 1 + (6.48 - 6.48i)T - 343iT^{2} \) |
| 11 | \( 1 + (-49.3 + 20.4i)T + (941. - 941. i)T^{2} \) |
| 13 | \( 1 + (-21.4 + 51.7i)T + (-1.55e3 - 1.55e3i)T^{2} \) |
| 17 | \( 1 + 3.73iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (36.7 - 88.6i)T + (-4.85e3 - 4.85e3i)T^{2} \) |
| 23 | \( 1 + (45.4 + 45.4i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + (-51.9 - 21.5i)T + (1.72e4 + 1.72e4i)T^{2} \) |
| 31 | \( 1 + 73.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + (165. + 399. i)T + (-3.58e4 + 3.58e4i)T^{2} \) |
| 41 | \( 1 + (-334. - 334. i)T + 6.89e4iT^{2} \) |
| 43 | \( 1 + (-328. + 136. i)T + (5.62e4 - 5.62e4i)T^{2} \) |
| 47 | \( 1 + 185. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (-412. + 171. i)T + (1.05e5 - 1.05e5i)T^{2} \) |
| 59 | \( 1 + (214. + 518. i)T + (-1.45e5 + 1.45e5i)T^{2} \) |
| 61 | \( 1 + (-85.1 - 35.2i)T + (1.60e5 + 1.60e5i)T^{2} \) |
| 67 | \( 1 + (252. + 104. i)T + (2.12e5 + 2.12e5i)T^{2} \) |
| 71 | \( 1 + (-430. + 430. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (-41.8 - 41.8i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 - 1.21e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (290. - 702. i)T + (-4.04e5 - 4.04e5i)T^{2} \) |
| 89 | \( 1 + (-365. + 365. i)T - 7.04e5iT^{2} \) |
| 97 | \( 1 - 508.T + 9.12e5T^{2} \) |
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
−16.22712710172602104213843275789, −14.33076272507482594823640461630, −12.62722067118835635107276061533, −12.22842333549906655386245685935, −10.99337882560958276445894828602, −9.224275356798787394487580415246, −8.332629157434290016514393379108, −5.77820694416192809280955622549, −3.80797267149740981085072869820, −0.75557527514635534891477809551,
4.20903388743908697347610181018, 6.32558210071226135514242247513, 7.20696092992906373472844546353, 9.040404772513349018412857247018, 10.52875270156405510411248039154, 11.65573535442990046649162519253, 13.74875248281394277942116699101, 14.61448061902420904843230413714, 15.75896218896751115663074647233, 16.84960908290817804711196106758