L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (2.20 − 0.344i)5-s + 0.999·6-s + (−2.24 − 1.40i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (2.08 + 0.806i)10-s + (−0.838 − 1.45i)11-s + (0.866 + 0.499i)12-s + 4.48i·13-s + (−1.24 − 2.33i)14-s + (1.74 − 1.40i)15-s + (−0.5 + 0.866i)16-s + (−6.59 + 3.80i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.988 − 0.153i)5-s + 0.408·6-s + (−0.847 − 0.530i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.659 + 0.255i)10-s + (−0.252 − 0.437i)11-s + (0.249 + 0.144i)12-s + 1.24i·13-s + (−0.331 − 0.624i)14-s + (0.449 − 0.362i)15-s + (−0.125 + 0.216i)16-s + (−1.59 + 0.922i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.96211 + 0.179715i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.96211 + 0.179715i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (-2.20 + 0.344i)T \) |
| 7 | \( 1 + (2.24 + 1.40i)T \) |
good | 11 | \( 1 + (0.838 + 1.45i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4.48iT - 13T^{2} \) |
| 17 | \( 1 + (6.59 - 3.80i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.24 + 3.88i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.417 - 0.241i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 1.19T + 29T^{2} \) |
| 31 | \( 1 + (-1.74 - 3.01i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.417 + 0.241i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + (9.91 + 5.72i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.29 + 4.78i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.88 - 4.99i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.28 + 9.15i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.90 + 1.67i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (-3.46 + 2i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.25 + 3.91i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 1.87iT - 83T^{2} \) |
| 89 | \( 1 + (1.67 - 2.90i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 17.7iT - 97T^{2} \) |
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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
−12.90929870555286408933292693290, −11.58822247753263435546872770969, −10.40717948913358895309657796773, −9.286678963913778858256956351217, −8.505101612697814121563461397780, −6.80100313182950560148903249680, −6.54156545833584443094925004290, −4.99034809272633362731590963618, −3.61916863167524711020739878430, −2.15010600216965396995730742929,
2.29474100624001875363625903247, 3.25119443692124351327819622914, 4.92283855199073762520618602148, 5.91516820848754337390625518045, 7.03102372957711629255533228191, 8.603228086228383543840260061982, 9.742302371863951459117182589298, 10.14875020537477914758023468190, 11.39991033695126373220654130360, 12.69427642521811978851118650349