| L(s) = 1 | + (−3.06 − 0.822i)2-s + (−0.866 − 1.5i)3-s + (5.27 + 3.04i)4-s + (−5.58 + 5.58i)5-s + (1.42 + 5.31i)6-s + (−7.30 + 1.95i)7-s + (−4.69 − 4.69i)8-s + (−1.5 + 2.59i)9-s + (21.7 − 12.5i)10-s + (3.01 − 11.2i)11-s − 10.5i·12-s + (−12.4 + 3.76i)13-s + 24.0·14-s + (13.2 + 3.54i)15-s + (−1.64 − 2.84i)16-s + (−10.4 − 6.05i)17-s + ⋯ |
| L(s) = 1 | + (−1.53 − 0.411i)2-s + (−0.288 − 0.5i)3-s + (1.31 + 0.761i)4-s + (−1.11 + 1.11i)5-s + (0.237 + 0.885i)6-s + (−1.04 + 0.279i)7-s + (−0.586 − 0.586i)8-s + (−0.166 + 0.288i)9-s + (2.17 − 1.25i)10-s + (0.273 − 1.02i)11-s − 0.878i·12-s + (−0.957 + 0.289i)13-s + 1.71·14-s + (0.880 + 0.236i)15-s + (−0.102 − 0.177i)16-s + (−0.616 − 0.355i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0289850 + 0.0705992i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0289850 + 0.0705992i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.866 + 1.5i)T \) |
| 13 | \( 1 + (12.4 - 3.76i)T \) |
| good | 2 | \( 1 + (3.06 + 0.822i)T + (3.46 + 2i)T^{2} \) |
| 5 | \( 1 + (5.58 - 5.58i)T - 25iT^{2} \) |
| 7 | \( 1 + (7.30 - 1.95i)T + (42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (-3.01 + 11.2i)T + (-104. - 60.5i)T^{2} \) |
| 17 | \( 1 + (10.4 + 6.05i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-2.73 - 10.1i)T + (-312. + 180.5i)T^{2} \) |
| 23 | \( 1 + (3.75 - 2.16i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-22.1 - 38.3i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (5.71 - 5.71i)T - 961iT^{2} \) |
| 37 | \( 1 + (-8.51 + 31.7i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + (58.1 + 15.5i)T + (1.45e3 + 840.5i)T^{2} \) |
| 43 | \( 1 + (-27.0 - 15.5i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-22.4 - 22.4i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + 54.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + (13.4 - 3.61i)T + (3.01e3 - 1.74e3i)T^{2} \) |
| 61 | \( 1 + (42.0 - 72.8i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-62.2 - 16.6i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + (14.8 + 55.2i)T + (-4.36e3 + 2.52e3i)T^{2} \) |
| 73 | \( 1 + (54.9 + 54.9i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 45.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + (30.8 - 30.8i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-5.84 + 21.8i)T + (-6.85e3 - 3.96e3i)T^{2} \) |
| 97 | \( 1 + (-28.7 - 107. i)T + (-8.14e3 + 4.70e3i)T^{2} \) |
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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
−16.58268476773702784657595686152, −15.81587279093646039121035124452, −14.21502869528273177463904661734, −12.28568097194641943733715356309, −11.37108637937078407181548901102, −10.41034729801860018604390602883, −8.981524414029644454120666917793, −7.58976592618003800528923665813, −6.63553457079320772278692858956, −2.99707337966433967388695074618,
0.13813031422891331898937600693, 4.45550352784539872604622467980, 6.81325059043780446003326912882, 8.080459581599900854868106794099, 9.324508772429738115742258004623, 10.12624399488937856683657660234, 11.73069550315070130959659423390, 12.86712695186996036520828718936, 15.34897999736201180379143925973, 15.79022515238476551096128492878
