| L(s) = 1 | + (−0.366 + 1.36i)2-s + (−2.15 + 0.578i)3-s + (−0.578 + 2.15i)5-s − 3.16i·6-s + (−2 + 1.99i)8-s + (1.73 − i)9-s + (−2.73 − 1.58i)10-s + (0.5 − 0.866i)11-s + (−1.58 − 1.58i)13-s − 5i·15-s + (−1.99 − 3.46i)16-s + (0.578 + 2.15i)17-s + (0.732 + 2.73i)18-s + (−1.58 − 2.73i)19-s + (0.999 + i)22-s + (−2.73 − 0.732i)23-s + ⋯ |
| L(s) = 1 | + (−0.258 + 0.965i)2-s + (−1.24 + 0.334i)3-s + (−0.258 + 0.965i)5-s − 1.29i·6-s + (−0.707 + 0.707i)8-s + (0.577 − 0.333i)9-s + (−0.866 − 0.499i)10-s + (0.150 − 0.261i)11-s + (−0.438 − 0.438i)13-s − 1.29i·15-s + (−0.499 − 0.866i)16-s + (0.140 + 0.523i)17-s + (0.172 + 0.643i)18-s + (−0.362 − 0.628i)19-s + (0.213 + 0.213i)22-s + (−0.569 − 0.152i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.730 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.730 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.163694 - 0.414645i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.163694 - 0.414645i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.578 - 2.15i)T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + (0.366 - 1.36i)T + (-1.73 - i)T^{2} \) |
| 3 | \( 1 + (2.15 - 0.578i)T + (2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.58 + 1.58i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.578 - 2.15i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (1.58 + 2.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.73 + 0.732i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 3iT - 29T^{2} \) |
| 31 | \( 1 + (-2.73 - 1.58i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.19 - 8.19i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 9.48iT - 41T^{2} \) |
| 43 | \( 1 + (3 - 3i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.47 - 1.73i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.366 - 1.36i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (4.74 - 8.21i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.47 - 3.16i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.36 + 0.366i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-11.2 + 6.5i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.16 + 3.16i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.16 - 5.47i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.58 - 1.58i)T - 97iT^{2} \) |
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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
−12.31524757275485780846161831647, −11.60112328178925434065055081516, −10.83795518054165488174009330101, −10.03514767537743803088965593021, −8.523562346714671886463091034987, −7.52033661865888494137912351344, −6.50033060611931061770330732910, −5.97155832637066978238970289406, −4.72624706192328025369538875064, −2.95259464493521119574422987255,
0.43147732704171169335970301677, 1.92987689536794835688552298703, 3.97180615654096607314888212198, 5.28916338903018807414013258628, 6.26757262518352928578981219749, 7.42538155024075231935437206479, 8.893019647940682898355455791159, 9.820971811269478986493534764109, 10.77342276886205386381177738254, 11.70038651314238492804994564821
