| L(s) = 1 | − 13.8i·2-s + 15.5·3-s − 126.·4-s + 76.4·5-s − 215. i·6-s − 5.71·7-s + 864. i·8-s + 243·9-s − 1.05e3i·10-s − 701. i·11-s − 1.97e3·12-s − 170. i·13-s + 78.9i·14-s + 1.19e3·15-s + 3.83e3·16-s − 6.75e3·17-s + ⋯ |
| L(s) = 1 | − 1.72i·2-s + 0.577·3-s − 1.97·4-s + 0.611·5-s − 0.996i·6-s − 0.0166·7-s + 1.68i·8-s + 0.333·9-s − 1.05i·10-s − 0.527i·11-s − 1.14·12-s − 0.0778i·13-s + 0.0287i·14-s + 0.353·15-s + 0.936·16-s − 1.37·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.120 - 0.992i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.040910643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.040910643\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 15.5T \) |
| 59 | \( 1 + (2.47e4 + 2.03e5i)T \) |
| good | 2 | \( 1 + 13.8iT - 64T^{2} \) |
| 5 | \( 1 - 76.4T + 1.56e4T^{2} \) |
| 7 | \( 1 + 5.71T + 1.17e5T^{2} \) |
| 11 | \( 1 + 701. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 170. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 6.75e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 1.48e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 2.31e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 3.09e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 1.61e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 6.68e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 5.02e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 6.28e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 7.30e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 3.79e3T + 2.21e10T^{2} \) |
| 61 | \( 1 - 2.75e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 2.29e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 4.05e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 1.56e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 4.12e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 9.55e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.87e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 7.06e5iT - 8.32e11T^{2} \) |
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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
−10.81788275662096027078500984558, −10.04979420985092634560857562118, −9.106880214275270174158286995014, −8.374389612301057833715017680614, −6.56993240177268503070501445462, −4.92217608991024201136701667824, −3.74749910526854584617885953865, −2.59859073947928840949829782186, −1.72941449407914766016395181040, −0.25302195846592677760032794501,
1.95169479414819484889968351632, 3.91395686109733735603205417791, 5.15066604082821434452361342640, 6.16037100435725369486831614010, 7.21045336795119521423582835219, 7.958128433151540619595426441765, 9.244745747873202878612160120090, 9.574877647992594579298528070824, 11.26653942554737997321273448603, 12.96865707560243075578614488795
