L(s) = 1 | + (−0.729 + 1.76i)3-s + (4.29 − 1.78i)5-s + (−1.47 + 1.47i)7-s + (16.5 + 16.5i)9-s + (−0.854 − 2.06i)11-s + (40.9 + 16.9i)13-s + 8.86i·15-s + 73.1i·17-s + (18.2 + 7.55i)19-s + (−1.52 − 3.68i)21-s + (144. + 144. i)23-s + (−73.0 + 73.0i)25-s + (−88.6 + 36.7i)27-s + (80.7 − 194. i)29-s + 168.·31-s + ⋯ |
L(s) = 1 | + (−0.140 + 0.338i)3-s + (0.384 − 0.159i)5-s + (−0.0798 + 0.0798i)7-s + (0.611 + 0.611i)9-s + (−0.0234 − 0.0565i)11-s + (0.874 + 0.362i)13-s + 0.152i·15-s + 1.04i·17-s + (0.220 + 0.0912i)19-s + (−0.0158 − 0.0382i)21-s + (1.30 + 1.30i)23-s + (−0.584 + 0.584i)25-s + (−0.632 + 0.261i)27-s + (0.517 − 1.24i)29-s + 0.978·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 - 0.767i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.54722 + 0.723088i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.54722 + 0.723088i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (0.729 - 1.76i)T + (-19.0 - 19.0i)T^{2} \) |
| 5 | \( 1 + (-4.29 + 1.78i)T + (88.3 - 88.3i)T^{2} \) |
| 7 | \( 1 + (1.47 - 1.47i)T - 343iT^{2} \) |
| 11 | \( 1 + (0.854 + 2.06i)T + (-941. + 941. i)T^{2} \) |
| 13 | \( 1 + (-40.9 - 16.9i)T + (1.55e3 + 1.55e3i)T^{2} \) |
| 17 | \( 1 - 73.1iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (-18.2 - 7.55i)T + (4.85e3 + 4.85e3i)T^{2} \) |
| 23 | \( 1 + (-144. - 144. i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + (-80.7 + 194. i)T + (-1.72e4 - 1.72e4i)T^{2} \) |
| 31 | \( 1 - 168.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (72.0 - 29.8i)T + (3.58e4 - 3.58e4i)T^{2} \) |
| 41 | \( 1 + (141. + 141. i)T + 6.89e4iT^{2} \) |
| 43 | \( 1 + (161. + 389. i)T + (-5.62e4 + 5.62e4i)T^{2} \) |
| 47 | \( 1 + 239. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (59.8 + 144. i)T + (-1.05e5 + 1.05e5i)T^{2} \) |
| 59 | \( 1 + (582. - 241. i)T + (1.45e5 - 1.45e5i)T^{2} \) |
| 61 | \( 1 + (-238. + 575. i)T + (-1.60e5 - 1.60e5i)T^{2} \) |
| 67 | \( 1 + (-156. + 376. i)T + (-2.12e5 - 2.12e5i)T^{2} \) |
| 71 | \( 1 + (411. - 411. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (642. + 642. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 800. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-1.34e3 - 557. i)T + (4.04e5 + 4.04e5i)T^{2} \) |
| 89 | \( 1 + (-340. + 340. i)T - 7.04e5iT^{2} \) |
| 97 | \( 1 - 632.T + 9.12e5T^{2} \) |
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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
−13.29334976303844323556819471285, −11.91421832935309866647286067588, −10.84978187607420285528820790715, −9.952235708932004339440637143689, −8.889832891225005454321289088682, −7.65322177824216107767195475683, −6.25818534225989944220685076847, −5.07712174498136307876370111322, −3.68862335266222295642234708180, −1.65092959512903089315916627287,
1.04153292066156965486272647081, 3.01556575419237639116126540584, 4.71815363273205143108506137628, 6.24828234514340362475912445237, 7.06220950477101865263180272491, 8.488959562312689293190693765794, 9.636760079368364537858056536505, 10.62902790891765098492477175379, 11.78549159246711436507301525038, 12.79692620804087101397052452390