L(s) = 1 | + (0.5 − 0.866i)3-s + 5-s + (−0.5 − 0.866i)7-s + (1 + 1.73i)9-s + (1.5 − 2.59i)11-s + (1 − 3.46i)13-s + (0.5 − 0.866i)15-s + (1.5 + 2.59i)17-s + (−3.5 − 6.06i)19-s − 0.999·21-s + (−1.5 + 2.59i)23-s + 25-s + 5·27-s + (−1.5 + 2.59i)29-s + 4·31-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + 0.447·5-s + (−0.188 − 0.327i)7-s + (0.333 + 0.577i)9-s + (0.452 − 0.783i)11-s + (0.277 − 0.960i)13-s + (0.129 − 0.223i)15-s + (0.363 + 0.630i)17-s + (−0.802 − 1.39i)19-s − 0.218·21-s + (−0.312 + 0.541i)23-s + 0.200·25-s + 0.962·27-s + (−0.278 + 0.482i)29-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.961434726\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.961434726\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + (-1 + 3.46i)T \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.5 - 9.52i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.5 + 2.59i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.5 - 6.06i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.756191588287871648859165811934, −8.877375824938135886173928856042, −8.097190056652044338575577282229, −7.34445580816877635323052172208, −6.39531123153605777100588934769, −5.64619761202281905132792913179, −4.49503836159037450786749216550, −3.34033797054344051392410717312, −2.26446068927483415855955850151, −0.953873267213028662058595664919,
1.51152303291633495478239196345, 2.73774811435100108151228557820, 4.01923353770038868914319732041, 4.56191153329794976121191230944, 5.99986547989200094985342036163, 6.50913462040832063426356666381, 7.58104929213287439766592975022, 8.658795893503019725909953996342, 9.380871773551121011846167388122, 9.874372311729542552209458190720