| L(s) = 1 | + 0.238i·2-s + 3.94·4-s − 9.50i·5-s − 5.89·7-s + 1.89i·8-s + 2.26·10-s − 9.02i·11-s + 3.60·13-s − 1.40i·14-s + 15.3·16-s + 19.8i·17-s + 32.2·19-s − 37.4i·20-s + 2.15·22-s + 2.25i·23-s + ⋯ |
| L(s) = 1 | + 0.119i·2-s + 0.985·4-s − 1.90i·5-s − 0.841·7-s + 0.236i·8-s + 0.226·10-s − 0.820i·11-s + 0.277·13-s − 0.100i·14-s + 0.957·16-s + 1.16i·17-s + 1.69·19-s − 1.87i·20-s + 0.0979·22-s + 0.0979i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.38544 - 0.717160i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38544 - 0.717160i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 - 3.60T \) |
| good | 2 | \( 1 - 0.238iT - 4T^{2} \) |
| 5 | \( 1 + 9.50iT - 25T^{2} \) |
| 7 | \( 1 + 5.89T + 49T^{2} \) |
| 11 | \( 1 + 9.02iT - 121T^{2} \) |
| 17 | \( 1 - 19.8iT - 289T^{2} \) |
| 19 | \( 1 - 32.2T + 361T^{2} \) |
| 23 | \( 1 - 2.25iT - 529T^{2} \) |
| 29 | \( 1 - 25.2iT - 841T^{2} \) |
| 31 | \( 1 - 8.89T + 961T^{2} \) |
| 37 | \( 1 + 38.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 29.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 45.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 51.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 47.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 33.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 10.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 63.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 92.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 57.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 22.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 108. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 65.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 36.4T + 9.40e3T^{2} \) |
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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
−12.90023385097438979478076787096, −12.27924270596526933540493086876, −11.22599180745249722846961280159, −9.841043083665339362324136062802, −8.778553014256907857273566996704, −7.80329384800721536923613602227, −6.23303052282266849810451865403, −5.28595421453125009501401212438, −3.47904714578236888857879846136, −1.25861746030581259933347928955,
2.51536723491900765715101839783, 3.41984835846068180050624208098, 5.89623770846633214502606235923, 7.02805613860950361381779579016, 7.37772863586674204017816350766, 9.688292356225533676325210242785, 10.31550876049810390027514280728, 11.38838646361895986573295703718, 12.07385673838523335962015292042, 13.62830241114388327200715978027
