| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 0.672i·5-s + 2.64·7-s + 2.82i·8-s + 0.951·10-s + 13.2i·11-s + 4.62·13-s − 3.74i·14-s + 4.00·16-s + 15.2i·17-s + 4.04·19-s − 1.34i·20-s + 18.7·22-s − 9.26i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 0.134i·5-s + 0.377·7-s + 0.353i·8-s + 0.0951·10-s + 1.20i·11-s + 0.356·13-s − 0.267i·14-s + 0.250·16-s + 0.894i·17-s + 0.212·19-s − 0.0672i·20-s + 0.852·22-s − 0.402i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.61861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61861\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 - 0.672iT - 25T^{2} \) |
| 11 | \( 1 - 13.2iT - 121T^{2} \) |
| 13 | \( 1 - 4.62T + 169T^{2} \) |
| 17 | \( 1 - 15.2iT - 289T^{2} \) |
| 19 | \( 1 - 4.04T + 361T^{2} \) |
| 23 | \( 1 + 9.26iT - 529T^{2} \) |
| 29 | \( 1 - 27.5iT - 841T^{2} \) |
| 31 | \( 1 - 44.0T + 961T^{2} \) |
| 37 | \( 1 + 2.90T + 1.36e3T^{2} \) |
| 41 | \( 1 + 23.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 62.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 55.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 1.84iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 13.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 20.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 75.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 91.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 84.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 80.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 109. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 126.T + 9.40e3T^{2} \) |
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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
−10.96628676789671279642681514866, −10.45527771315697234263009644991, −9.432602809942715293615606234953, −8.521655461060109453251291824346, −7.50102394232686234679842863433, −6.34896721942338990248888864680, −4.99453624296871308658950061311, −4.09742752143146985357131842752, −2.69227661645137965654041865725, −1.36824445190420871863605761703,
0.847775297730941222290929182547, 2.96245858749730092027193178325, 4.34676465417062552500811679599, 5.43031922937342308540830219455, 6.31250789483230082887678300144, 7.42747539174422483914789672235, 8.351121418080772612512647699331, 9.045739892475111813096473947387, 10.14186849196782273143036298470, 11.21773329863109427824924280106
