| L(s) = 1 | + i·3-s + (0.726 + 2.11i)5-s + 4.05i·7-s − 9-s − 0.985·11-s + 4.94i·13-s + (−2.11 + 0.726i)15-s − 4.52i·17-s + 2.60·19-s − 4.05·21-s + 3.53i·23-s + (−3.94 + 3.07i)25-s − i·27-s − 7.59·29-s + 3.28·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.324 + 0.945i)5-s + 1.53i·7-s − 0.333·9-s − 0.297·11-s + 1.37i·13-s + (−0.546 + 0.187i)15-s − 1.09i·17-s + 0.597·19-s − 0.885·21-s + 0.737i·23-s + (−0.789 + 0.614i)25-s − 0.192i·27-s − 1.41·29-s + 0.589·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.450376169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.450376169\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-0.726 - 2.11i)T \) |
| good | 7 | \( 1 - 4.05iT - 7T^{2} \) |
| 11 | \( 1 + 0.985T + 11T^{2} \) |
| 13 | \( 1 - 4.94iT - 13T^{2} \) |
| 17 | \( 1 + 4.52iT - 17T^{2} \) |
| 19 | \( 1 - 2.60T + 19T^{2} \) |
| 23 | \( 1 - 3.53iT - 23T^{2} \) |
| 29 | \( 1 + 7.59T + 29T^{2} \) |
| 31 | \( 1 - 3.28T + 31T^{2} \) |
| 37 | \( 1 - 0.945iT - 37T^{2} \) |
| 41 | \( 1 + 0.568T + 41T^{2} \) |
| 43 | \( 1 - 8.45iT - 43T^{2} \) |
| 47 | \( 1 - 2.60iT - 47T^{2} \) |
| 53 | \( 1 + 0.229iT - 53T^{2} \) |
| 59 | \( 1 - 9.10T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 8.45iT - 67T^{2} \) |
| 71 | \( 1 - 1.43T + 71T^{2} \) |
| 73 | \( 1 + 11.9iT - 73T^{2} \) |
| 79 | \( 1 + 3.28T + 79T^{2} \) |
| 83 | \( 1 + 9.89iT - 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 3.23iT - 97T^{2} \) |
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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
−9.205544913983984802652024655585, −8.252717637800781107082760563829, −7.33178197132847338131551940543, −6.66025337939894173618595956566, −5.76503083911146983663541242683, −5.34050171825446221499848486276, −4.36397645222477963875909546588, −3.28451771030675875295399660459, −2.62836170265609497336713140517, −1.83167741227152665586215807798,
0.44019877344884234986424399094, 1.16251886046199543665830978563, 2.25325698036568601254318002417, 3.53301665770179419940230042377, 4.16406034776814110893979525126, 5.24725594922556452177904486752, 5.71218196983436870801053712789, 6.73255953110393741048318244209, 7.41195460511340466306520734474, 8.134877499331414046818960180942
