| L(s) = 1 | + (−0.913 + 0.406i)2-s + (0.978 − 0.207i)3-s + (0.669 − 0.743i)4-s + (−0.126 − 1.19i)5-s + (−0.809 + 0.587i)6-s + (0.836 + 2.50i)7-s + (−0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s + (0.603 + 1.04i)10-s + (−2.32 + 2.36i)11-s + (0.499 − 0.866i)12-s + (3.40 + 2.47i)13-s + (−1.78 − 1.95i)14-s + (−0.372 − 1.14i)15-s + (−0.104 − 0.994i)16-s + (3.96 + 1.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.645 + 0.287i)2-s + (0.564 − 0.120i)3-s + (0.334 − 0.371i)4-s + (−0.0564 − 0.536i)5-s + (−0.330 + 0.239i)6-s + (0.316 + 0.948i)7-s + (−0.109 + 0.336i)8-s + (0.304 − 0.135i)9-s + (0.190 + 0.330i)10-s + (−0.699 + 0.714i)11-s + (0.144 − 0.249i)12-s + (0.943 + 0.685i)13-s + (−0.477 − 0.521i)14-s + (−0.0962 − 0.296i)15-s + (−0.0261 − 0.248i)16-s + (0.960 + 0.427i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26307 + 0.374302i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26307 + 0.374302i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.913 - 0.406i)T \) |
| 3 | \( 1 + (-0.978 + 0.207i)T \) |
| 7 | \( 1 + (-0.836 - 2.50i)T \) |
| 11 | \( 1 + (2.32 - 2.36i)T \) |
| good | 5 | \( 1 + (0.126 + 1.19i)T + (-4.89 + 1.03i)T^{2} \) |
| 13 | \( 1 + (-3.40 - 2.47i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.96 - 1.76i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (0.0175 + 0.0194i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-2.68 + 4.64i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.918 + 2.82i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.281 - 2.67i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-6.33 - 1.34i)T + (33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (-0.792 + 2.44i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 1.01T + 43T^{2} \) |
| 47 | \( 1 + (7.61 + 8.46i)T + (-4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (1.23 - 11.7i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (1.43 - 1.59i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.710 - 6.76i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (1.64 + 2.84i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.42 - 5.39i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.86 + 2.07i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-11.4 + 5.11i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (1.92 - 1.39i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.52 + 2.64i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (15.2 + 11.0i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.98955633323512056544370898206, −10.02947376440053680824928268409, −9.072156407573543148895937198172, −8.504724778914029016112413394643, −7.77971045206497934624139245870, −6.63048644118787727905612507732, −5.55101772088664901895086387307, −4.45727127817519261196991977360, −2.77594735266064316123571965841, −1.50670802816257399491345670164,
1.13543714287573597012692899192, 2.96470479458824487775717491620, 3.62472925860898779587783203028, 5.22102575369780274499767434264, 6.58512917996249984558875052498, 7.74492436766826874368815438470, 8.034560591741871549698708185948, 9.263371550739459655692146575517, 10.12966653079729655877796273073, 10.93310031917398511675133550572
