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 | \( 1 - 508.T + 9.12e5T^{2} \) |
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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
−16.84960908290817804711196106758, −15.75896218896751115663074647233, −14.61448061902420904843230413714, −13.74875248281394277942116699101, −11.65573535442990046649162519253, −10.52875270156405510411248039154, −9.040404772513349018412857247018, −7.20696092992906373472844546353, −6.32558210071226135514242247513, −4.20903388743908697347610181018,
0.75557527514635534891477809551, 3.80797267149740981085072869820, 5.77820694416192809280955622549, 8.332629157434290016514393379108, 9.224275356798787394487580415246, 10.99337882560958276445894828602, 12.22842333549906655386245685935, 12.62722067118835635107276061533, 14.33076272507482594823640461630, 16.22712710172602104213843275789