| L(s) = 1 | − 4i·2-s + 2.52i·3-s − 16·4-s + 10.1·6-s + 49i·7-s + 64i·8-s + 236.·9-s − 267.·11-s − 40.4i·12-s + 896. i·13-s + 196·14-s + 256·16-s − 61.1i·17-s − 946. i·18-s − 1.62e3·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.162i·3-s − 0.5·4-s + 0.114·6-s + 0.377i·7-s + 0.353i·8-s + 0.973·9-s − 0.667·11-s − 0.0811i·12-s + 1.47i·13-s + 0.267·14-s + 0.250·16-s − 0.0513i·17-s − 0.688i·18-s − 1.03·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.03239791260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03239791260\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 49iT \) |
| good | 3 | \( 1 - 2.52iT - 243T^{2} \) |
| 11 | \( 1 + 267.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 896. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 61.1iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.62e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.28e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 7.42e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.93e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 640. iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 3.87e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.97e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.06e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.97e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.64e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.46e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.40e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 1.99e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.05e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 6.03e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.21e5iT - 8.58e9T^{2} \) |
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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
−10.81361636569777055084295381098, −10.37988969964562802465597432019, −9.230367050323480989993433787266, −8.589436791201945474808697789892, −7.26430957855566921337318736143, −6.23704421236898820378142822265, −4.76051638181626058174911386654, −4.13580239737364879224118844548, −2.61122067145111493156955462707, −1.62335900851171546268618521582,
0.008268051112501743719563215198, 1.39181402855681858559124210733, 3.13974886455495503290148152174, 4.40192015873810799369049840029, 5.42269404384669105920751241472, 6.46060282983758481840909138890, 7.58111648398833761636345980237, 7.986539443461249917783059922530, 9.313562648503002847283029563488, 10.24622599385282984823748174269
