| L(s) = 1 | + 1.41i·2-s − 2.00·4-s + 4.24i·5-s + 4·7-s − 2.82i·8-s − 6·10-s − 8·13-s + 5.65i·14-s + 4.00·16-s − 12.7i·17-s + 16·19-s − 8.48i·20-s + 16.9i·23-s + 7.00·25-s − 11.3i·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 0.848i·5-s + 0.571·7-s − 0.353i·8-s − 0.600·10-s − 0.615·13-s + 0.404i·14-s + 0.250·16-s − 0.748i·17-s + 0.842·19-s − 0.424i·20-s + 0.737i·23-s + 0.280·25-s − 0.435i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.751891451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.751891451\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 4.24iT - 25T^{2} \) |
| 7 | \( 1 - 4T + 49T^{2} \) |
| 13 | \( 1 + 8T + 169T^{2} \) |
| 17 | \( 1 + 12.7iT - 289T^{2} \) |
| 19 | \( 1 - 16T + 361T^{2} \) |
| 23 | \( 1 - 16.9iT - 529T^{2} \) |
| 29 | \( 1 - 4.24iT - 841T^{2} \) |
| 31 | \( 1 - 44T + 961T^{2} \) |
| 37 | \( 1 + 34T + 1.36e3T^{2} \) |
| 41 | \( 1 - 46.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40T + 1.84e3T^{2} \) |
| 47 | \( 1 - 84.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 38.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 50T + 3.72e3T^{2} \) |
| 67 | \( 1 - 8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 16T + 5.32e3T^{2} \) |
| 79 | \( 1 - 76T + 6.24e3T^{2} \) |
| 83 | \( 1 - 118. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 12.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 176T + 9.40e3T^{2} \) |
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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
−9.254426823772443343308504846960, −8.132746065482313875655401861765, −7.61785930766991125729281594995, −6.91412094718681634192684719609, −6.21619131269084257003661779650, −5.18058867250044283094242515016, −4.64648888447975387667374644830, −3.39537073467196644972500432557, −2.58396627154243996818080489268, −1.11684891718331681338216124601,
0.49578290399755376893829270373, 1.50195170149291168393819821005, 2.48889779568035641654487503192, 3.62087637449526051174542915430, 4.61288117831007454162491419490, 5.06652066419139484113551489509, 6.02794642958034708679515308015, 7.17000475175418607505028815339, 8.046761744815385870946715381597, 8.676260667569019795823993990636
