| L(s) = 1 | + (−2.73 − 2.73i)2-s + 11i·4-s + (19.1 − 19.1i)8-s − 61.0·16-s + (−21.9 − 21.9i)17-s − 22i·19-s + (−21.9 + 21.9i)23-s − 2·31-s + (90.3 + 90.3i)32-s + 120i·34-s + (−60.2 + 60.2i)38-s + 120·46-s + (−65.7 − 65.7i)47-s − 49i·49-s + (−43.8 + 43.8i)53-s + ⋯ |
| L(s) = 1 | + (−1.36 − 1.36i)2-s + 2.75i·4-s + (2.39 − 2.39i)8-s − 3.81·16-s + (−1.28 − 1.28i)17-s − 1.15i·19-s + (−0.952 + 0.952i)23-s − 0.0645·31-s + (2.82 + 2.82i)32-s + 3.52i·34-s + (−1.58 + 1.58i)38-s + 2.60·46-s + (−1.39 − 1.39i)47-s − 0.999i·49-s + (−0.826 + 0.826i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0755574 + 0.265973i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0755574 + 0.265973i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (2.73 + 2.73i)T + 4iT^{2} \) |
| 7 | \( 1 + 49iT^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 - 169iT^{2} \) |
| 17 | \( 1 + (21.9 + 21.9i)T + 289iT^{2} \) |
| 19 | \( 1 + 22iT - 361T^{2} \) |
| 23 | \( 1 + (21.9 - 21.9i)T - 529iT^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 2T + 961T^{2} \) |
| 37 | \( 1 + 1.36e3iT^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3iT^{2} \) |
| 47 | \( 1 + (65.7 + 65.7i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (43.8 - 43.8i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 118T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3iT^{2} \) |
| 71 | \( 1 + 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3iT^{2} \) |
| 79 | \( 1 + 98iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-43.8 + 43.8i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 9.40e3iT^{2} \) |
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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
−11.41318578030887652232947769473, −10.48379050724398933041505168053, −9.476875658822184536598966549656, −8.905540783836899547859258364014, −7.78083339993714620212889445127, −6.83470359526966996698559614028, −4.61530659453028256684999096811, −3.19714912121795200649681853331, −1.99420629929663501466532435632, −0.22981660182098228922443903012,
1.74148456731391894367844642438, 4.49388909844363532316251117371, 5.97385339340510507933544251691, 6.53662066308075619474913943166, 7.85132226455557793632681036504, 8.423910199680404325616157462590, 9.439126538329183665826433573144, 10.34896071681261961078617710142, 11.08217044335563741153982608653, 12.64688811885727210081439554634
