| L(s) = 1 | + (0.939 + 0.342i)2-s + (1.72 − 0.104i)3-s + (0.766 + 0.642i)4-s + (−0.163 + 0.927i)5-s + (1.66 + 0.493i)6-s + (−0.766 + 0.642i)7-s + (0.500 + 0.866i)8-s + (2.97 − 0.359i)9-s + (−0.471 + 0.815i)10-s + (−0.205 − 1.16i)11-s + (1.39 + 1.03i)12-s + (0.528 − 0.192i)13-s + (−0.939 + 0.342i)14-s + (−0.186 + 1.62i)15-s + (0.173 + 0.984i)16-s + (−1.22 + 2.12i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.998 − 0.0600i)3-s + (0.383 + 0.321i)4-s + (−0.0731 + 0.414i)5-s + (0.677 + 0.201i)6-s + (−0.289 + 0.242i)7-s + (0.176 + 0.306i)8-s + (0.992 − 0.119i)9-s + (−0.148 + 0.258i)10-s + (−0.0619 − 0.351i)11-s + (0.401 + 0.297i)12-s + (0.146 − 0.0533i)13-s + (−0.251 + 0.0914i)14-s + (−0.0481 + 0.418i)15-s + (0.0434 + 0.246i)16-s + (−0.297 + 0.514i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.46005 + 0.744185i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.46005 + 0.744185i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 3 | \( 1 + (-1.72 + 0.104i)T \) |
| 7 | \( 1 + (0.766 - 0.642i)T \) |
| good | 5 | \( 1 + (0.163 - 0.927i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (0.205 + 1.16i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.528 + 0.192i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.22 - 2.12i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.07 + 3.60i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.572 - 0.480i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (9.48 + 3.45i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (3.33 + 2.79i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-2.75 + 4.77i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.559 - 0.203i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.948 + 5.38i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (5.74 - 4.81i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 4.38T + 53T^{2} \) |
| 59 | \( 1 + (0.408 - 2.31i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (4.53 - 3.80i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-6.83 + 2.48i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (3.42 - 5.92i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.138 - 0.239i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.48 - 1.26i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (12.9 + 4.72i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-0.814 - 1.41i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.961 - 5.45i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.44268842992246482904702836777, −10.66202909332456205718462066356, −9.423817326137284211643653532271, −8.639994928918639032396267965959, −7.58444168157424619650059498393, −6.78626510151412455040703104013, −5.68920317826305099997721215856, −4.25058617085126345570199677055, −3.30281899833711317385395895693, −2.20395229461431572878997140238,
1.73755178557756056260029425399, 3.12906956117088027493360420733, 4.12304896433201392312223606608, 5.12297299870816811613961541653, 6.57807480307902338729621925152, 7.50035214280732794877542361041, 8.570459794314761501300787354589, 9.472925287924142253486744190875, 10.31931528609697377275594470876, 11.32175767711578509399304905486
