| L(s) = 1 | + (0.939 − 0.342i)2-s + (−0.0350 − 1.73i)3-s + (0.766 − 0.642i)4-s + (−0.00462 − 0.0262i)5-s + (−0.625 − 1.61i)6-s + (−0.766 − 0.642i)7-s + (0.500 − 0.866i)8-s + (−2.99 + 0.121i)9-s + (−0.0133 − 0.0230i)10-s + (0.928 − 5.26i)11-s + (−1.13 − 1.30i)12-s + (−2.00 − 0.730i)13-s + (−0.939 − 0.342i)14-s + (−0.0452 + 0.00892i)15-s + (0.173 − 0.984i)16-s + (3.97 + 6.88i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.0202 − 0.999i)3-s + (0.383 − 0.321i)4-s + (−0.00206 − 0.0117i)5-s + (−0.255 − 0.659i)6-s + (−0.289 − 0.242i)7-s + (0.176 − 0.306i)8-s + (−0.999 + 0.0404i)9-s + (−0.00420 − 0.00728i)10-s + (0.280 − 1.58i)11-s + (−0.329 − 0.376i)12-s + (−0.556 − 0.202i)13-s + (−0.251 − 0.0914i)14-s + (−0.0116 + 0.00230i)15-s + (0.0434 − 0.246i)16-s + (0.964 + 1.67i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.339 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04470 - 1.48794i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04470 - 1.48794i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.0350 + 1.73i)T \) |
| 7 | \( 1 + (0.766 + 0.642i)T \) |
| good | 5 | \( 1 + (0.00462 + 0.0262i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.928 + 5.26i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (2.00 + 0.730i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.97 - 6.88i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.24 + 2.15i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.588 + 0.493i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (5.00 - 1.82i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (1.09 - 0.921i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (2.77 + 4.81i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-9.08 - 3.30i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.277 - 1.57i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.11 - 4.29i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 6.91T + 53T^{2} \) |
| 59 | \( 1 + (-1.56 - 8.85i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (3.41 + 2.86i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.133 - 0.0487i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.49 - 4.31i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.79 - 8.30i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-15.9 + 5.79i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.84 + 2.49i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (5.05 - 8.75i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.308 - 1.74i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.14632394237402281382781254837, −10.60229977725713933853032474447, −9.133755347823829527199166239775, −8.154883686302156459133549341253, −7.17449150042185616255644545165, −6.12091071205080700956086978164, −5.49968752323777222849942712762, −3.77638040230085279969527164121, −2.76136803019281740335232326617, −1.07329099256279366057092107733,
2.52083019040227070184823872411, 3.73267424995293724220313509996, 4.84468188512131987426314549768, 5.47453783631111066857434801732, 6.89116109433623900757116069819, 7.69234851266603092509014854327, 9.265034908105741767924422655835, 9.663493808510765738375130823441, 10.71651941491121140004707679388, 11.94161979975094602396720617161
