| L(s) = 1 | + (2.82 − 0.497i)2-s + (−2.53 − 3.02i)3-s + (3.95 − 1.44i)4-s + (−4.36 − 1.58i)5-s + (−8.66 − 7.27i)6-s + (−5.59 + 9.68i)7-s + (0.530 − 0.306i)8-s + (−1.14 + 6.47i)9-s + (−13.1 − 2.31i)10-s + (6.36 + 11.0i)11-s + (−14.4 − 8.31i)12-s + (3.10 − 3.69i)13-s + (−10.9 + 30.1i)14-s + (6.27 + 17.2i)15-s + (−11.5 + 9.70i)16-s + (−1.93 − 10.9i)17-s + ⋯ |
| L(s) = 1 | + (1.41 − 0.248i)2-s + (−0.845 − 1.00i)3-s + (0.989 − 0.360i)4-s + (−0.873 − 0.317i)5-s + (−1.44 − 1.21i)6-s + (−0.799 + 1.38i)7-s + (0.0663 − 0.0383i)8-s + (−0.126 + 0.719i)9-s + (−1.31 − 0.231i)10-s + (0.578 + 1.00i)11-s + (−1.20 − 0.692i)12-s + (0.238 − 0.284i)13-s + (−0.783 + 2.15i)14-s + (0.418 + 1.14i)15-s + (−0.722 + 0.606i)16-s + (−0.113 − 0.644i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.264171 + 0.358832i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.264171 + 0.358832i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (-2.82 + 0.497i)T + (3.75 - 1.36i)T^{2} \) |
| 3 | \( 1 + (2.53 + 3.02i)T + (-1.56 + 8.86i)T^{2} \) |
| 5 | \( 1 + (4.36 + 1.58i)T + (19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (5.59 - 9.68i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-6.36 - 11.0i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-3.10 + 3.69i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (1.93 + 10.9i)T + (-271. + 98.8i)T^{2} \) |
| 23 | \( 1 + (24.4 - 8.90i)T + (405. - 340. i)T^{2} \) |
| 29 | \( 1 + (3.55 + 0.626i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (6.73 + 3.88i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 20.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-5.13 - 6.12i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (49.1 + 17.8i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-1.48 + 8.43i)T + (-2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + (2.56 + 7.04i)T + (-2.15e3 + 1.80e3i)T^{2} \) |
| 59 | \( 1 + (59.0 - 10.4i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (81.3 - 29.6i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-125. - 22.0i)T + (4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (43.2 - 118. i)T + (-3.86e3 - 3.24e3i)T^{2} \) |
| 73 | \( 1 + (-36.8 + 30.9i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (52.1 + 62.1i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-48.5 + 84.1i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-2.40 + 2.86i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-83.7 + 14.7i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.88929184625207970184781577420, −11.43701674862626913953965483956, −9.731045241473809347509226538843, −8.605650759746336727089968930411, −7.30434857160537838510432879679, −6.33096436214973300929149188204, −5.69908781011472350436748840982, −4.62975242188365396663111146425, −3.41310612612834197665519514416, −2.02801050196325305996180157220,
0.13423086788758110457467819328, 3.65690530399483454695434302087, 3.73340330390738398439574160157, 4.72684806394898643670062895129, 6.03182515282735014793251643548, 6.58015224548041157234855803813, 7.78706653743682224190229120447, 9.356395874825314185761706875664, 10.43137049206335998736371918921, 11.10528042924259205061521062769
