L(s) = 1 | + (0.5 + 0.866i)3-s + (1.33 + 2.31i)5-s + 3.67·7-s + (−0.499 + 0.866i)9-s + 3.81·11-s + (−0.0719 + 0.124i)13-s + (−1.33 + 2.31i)15-s + (−4.24 + 0.990i)19-s + (1.83 + 3.18i)21-s + (3.76 − 6.52i)23-s + (−1.07 + 1.85i)25-s − 0.999·27-s + (2.67 − 4.62i)29-s − 8.81·31-s + (1.90 + 3.30i)33-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (0.597 + 1.03i)5-s + 1.38·7-s + (−0.166 + 0.288i)9-s + 1.15·11-s + (−0.0199 + 0.0345i)13-s + (−0.345 + 0.597i)15-s + (−0.973 + 0.227i)19-s + (0.400 + 0.694i)21-s + (0.784 − 1.35i)23-s + (−0.214 + 0.371i)25-s − 0.192·27-s + (0.496 − 0.859i)29-s − 1.58·31-s + (0.332 + 0.575i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.98943 + 1.17050i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.98943 + 1.17050i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (4.24 - 0.990i)T \) |
good | 5 | \( 1 + (-1.33 - 2.31i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 3.67T + 7T^{2} \) |
| 11 | \( 1 - 3.81T + 11T^{2} \) |
| 13 | \( 1 + (0.0719 - 0.124i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.76 + 6.52i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.67 + 4.62i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 8.81T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (2.67 + 4.62i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.40 - 2.43i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.00 - 6.94i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.90 - 3.30i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.74 - 9.95i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.69 + 4.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.81 + 11.8i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.172 - 0.299i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.26 - 5.65i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.28T + 83T^{2} \) |
| 89 | \( 1 + (-4.33 + 7.51i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.95 - 5.12i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.52816180307033288232081455851, −9.321136161266522709633899826947, −8.695595420410064006317286560542, −7.75936140975468510766284008072, −6.75619879491873938563068586270, −5.99866644849616802944813079689, −4.78496160871144732162276814711, −4.01358151081215318121851385750, −2.70808361790408769566856713064, −1.70398043451543250361502145971,
1.31163375389445785080215730663, 1.87335476968152249903907577889, 3.61896890045984704043603794984, 4.80860422408671284565946724250, 5.41107166601271901234553245980, 6.58622649637564513378121975557, 7.48045977610309180578869957956, 8.503103355968150263114250909784, 8.898184799462196020115279225251, 9.687602445260114606273962647418