| L(s) = 1 | + (1.27 − 3.92i)2-s + (−4.69 − 2.22i)3-s + (−7.33 − 5.32i)4-s + (−0.616 + 0.200i)5-s + (−14.7 + 15.6i)6-s + (8.50 − 11.7i)7-s + (−3.54 + 2.57i)8-s + (17.1 + 20.8i)9-s + 2.67i·10-s + (−16.7 − 32.3i)11-s + (22.5 + 41.3i)12-s + (54.2 + 17.6i)13-s + (−35.1 − 48.3i)14-s + (3.34 + 0.430i)15-s + (−16.8 − 51.7i)16-s + (5.39 + 16.5i)17-s + ⋯ |
| L(s) = 1 | + (0.451 − 1.38i)2-s + (−0.903 − 0.428i)3-s + (−0.916 − 0.665i)4-s + (−0.0551 + 0.0179i)5-s + (−1.00 + 1.06i)6-s + (0.459 − 0.632i)7-s + (−0.156 + 0.113i)8-s + (0.633 + 0.773i)9-s + 0.0846i·10-s + (−0.459 − 0.887i)11-s + (0.543 + 0.993i)12-s + (1.15 + 0.376i)13-s + (−0.670 − 0.923i)14-s + (0.0575 + 0.00740i)15-s + (−0.262 − 0.808i)16-s + (0.0769 + 0.236i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.598i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.383705 - 1.15417i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.383705 - 1.15417i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.69 + 2.22i)T \) |
| 11 | \( 1 + (16.7 + 32.3i)T \) |
| good | 2 | \( 1 + (-1.27 + 3.92i)T + (-6.47 - 4.70i)T^{2} \) |
| 5 | \( 1 + (0.616 - 0.200i)T + (101. - 73.4i)T^{2} \) |
| 7 | \( 1 + (-8.50 + 11.7i)T + (-105. - 326. i)T^{2} \) |
| 13 | \( 1 + (-54.2 - 17.6i)T + (1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-5.39 - 16.5i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-62.1 - 85.4i)T + (-2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 - 145. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (26.1 + 19.0i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-80.1 + 246. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-120. - 87.7i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (246. - 179. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 267. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-147. - 203. i)T + (-3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-225. - 73.3i)T + (1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-394. + 543. i)T + (-6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-188. + 61.1i)T + (1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + 557.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (226. - 73.6i)T + (2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (420. - 578. i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (973. + 316. i)T + (3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-278. - 857. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 - 165. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-158. + 487. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−15.94674628750613443853791607117, −13.76099106684046774082265884746, −13.24597483961395154005473316523, −11.67939690249311391444839376310, −11.20670609899165266886874153707, −10.04192095891675455891942456505, −7.72931316845869796794454745936, −5.69556525983933957225746701827, −3.87081202419310846815795548337, −1.31796909105367240486843224134,
4.63440670483614303700339244542, 5.70895277401103453967427495304, 7.03244147037581458734092904150, 8.628917112173898464682136157101, 10.48998903740678587842644682565, 11.90601743766058993799690592333, 13.30076194758663673051510946048, 14.79341031508795825862316608889, 15.66200188776173958787122174898, 16.28708523285210559214114731569
