| L(s) = 1 | + (1.36 − 0.359i)2-s + (−0.766 + 0.642i)3-s + (1.74 − 0.983i)4-s + (0.0388 + 0.0141i)5-s + (−0.816 + 1.15i)6-s + (1.39 + 0.804i)7-s + (2.02 − 1.97i)8-s + (0.173 − 0.984i)9-s + (0.0582 + 0.00537i)10-s + (3.14 − 1.81i)11-s + (−0.701 + 1.87i)12-s + (−2.36 + 2.81i)13-s + (2.19 + 0.599i)14-s + (−0.0388 + 0.0141i)15-s + (2.06 − 3.42i)16-s + (0.383 + 2.17i)17-s + ⋯ |
| L(s) = 1 | + (0.967 − 0.254i)2-s + (−0.442 + 0.371i)3-s + (0.870 − 0.491i)4-s + (0.0173 + 0.00632i)5-s + (−0.333 + 0.471i)6-s + (0.526 + 0.304i)7-s + (0.717 − 0.697i)8-s + (0.0578 − 0.328i)9-s + (0.0184 + 0.00169i)10-s + (0.946 − 0.546i)11-s + (−0.202 + 0.540i)12-s + (−0.655 + 0.781i)13-s + (0.586 + 0.160i)14-s + (−0.0100 + 0.00365i)15-s + (0.516 − 0.856i)16-s + (0.0929 + 0.527i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.95493 - 0.140077i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.95493 - 0.140077i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 + 0.359i)T \) |
| 3 | \( 1 + (0.766 - 0.642i)T \) |
| 19 | \( 1 + (3.73 - 2.24i)T \) |
| good | 5 | \( 1 + (-0.0388 - 0.0141i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.39 - 0.804i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.14 + 1.81i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.36 - 2.81i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.383 - 2.17i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (1.81 + 4.98i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (6.31 + 1.11i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (1.55 - 2.69i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.68iT - 37T^{2} \) |
| 41 | \( 1 + (-1.14 - 1.36i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (1.47 - 4.05i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.75 - 0.308i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-0.641 - 1.76i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (2.50 + 14.1i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (2.28 - 0.831i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (2.50 - 14.2i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-13.1 - 4.79i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-6.09 + 5.11i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (2.73 - 2.29i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.224 + 0.129i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.77 + 5.69i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-14.1 + 2.50i)T + (91.1 - 33.1i)T^{2} \) |
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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
−12.05460516931677609220974352258, −11.49065996467323666782342826670, −10.56370308826032932221527286130, −9.545212252334684324657565571823, −8.216295888363220176423254486153, −6.68185340951108465359112437455, −5.92751763358618075480432590075, −4.67859240699742374629488265444, −3.79642889976653214897416566497, −1.96375176697825638781924951789,
1.98326539142095316277523317977, 3.80407467109819296600179565897, 4.97726032438112105817394207357, 5.94643255284870481419650469582, 7.19568736133267354233372900264, 7.72673329751751457927677513598, 9.351757551444952747480876641022, 10.72827316344665241914884763891, 11.53146071099952646107750164937, 12.30020506879775133867245983892
