| L(s) = 1 | + (−0.951 + 0.309i)2-s + (−0.361 − 1.69i)3-s + (0.809 − 0.587i)4-s + (0.866 − 0.5i)5-s + (0.867 + 1.49i)6-s + (0.291 + 2.77i)7-s + (−0.587 + 0.809i)8-s + (−2.73 + 1.22i)9-s + (−0.669 + 0.743i)10-s + (0.844 + 0.376i)11-s + (−1.28 − 1.15i)12-s + (0.183 + 0.864i)13-s + (−1.13 − 2.54i)14-s + (−1.16 − 1.28i)15-s + (0.309 − 0.951i)16-s + (−6.74 + 3.00i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (−0.208 − 0.977i)3-s + (0.404 − 0.293i)4-s + (0.387 − 0.223i)5-s + (0.354 + 0.611i)6-s + (0.110 + 1.04i)7-s + (−0.207 + 0.286i)8-s + (−0.912 + 0.408i)9-s + (−0.211 + 0.235i)10-s + (0.254 + 0.113i)11-s + (−0.371 − 0.334i)12-s + (0.0509 + 0.239i)13-s + (−0.302 − 0.680i)14-s + (−0.299 − 0.332i)15-s + (0.0772 − 0.237i)16-s + (−1.63 + 0.728i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 - 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0170997 + 0.0671369i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0170997 + 0.0671369i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (0.361 + 1.69i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 + (5.55 + 0.339i)T \) |
| good | 7 | \( 1 + (-0.291 - 2.77i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-0.844 - 0.376i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-0.183 - 0.864i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (6.74 - 3.00i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (4.78 + 1.01i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (4.70 + 3.41i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.07 + 6.38i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (5.69 + 3.28i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.51 + 4.96i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (0.327 - 1.54i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-8.09 - 2.62i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (1.16 - 11.1i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (10.2 - 9.20i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 3.49iT - 61T^{2} \) |
| 67 | \( 1 + (1.98 + 3.43i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-15.9 - 1.67i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (0.0562 - 0.126i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (4.01 + 9.01i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (11.6 - 12.9i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-9.03 + 6.56i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (10.3 - 7.52i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.53967927140782297457496209251, −9.140870697140800318291097116596, −8.809494535554347647248406510775, −8.046737546362573887359279558777, −6.95651782869317805673602835962, −6.21035427412831174138182219258, −5.67720349784284330702874544475, −4.31851409579303546983022698324, −2.32296204549287303760470697776, −1.90851710844036989405867102601,
0.03781764987592042269075437730, 1.92403487894143219473768892013, 3.36394119146645035155671520246, 4.18664807116290100840128103702, 5.24871271649564674236262655267, 6.44263631692461732704697462514, 7.11340184324851651219402054654, 8.328143182339545707469170628476, 9.041759215708945735023127073962, 9.812965997244063881209189728254
