L(s) = 1 | + (0.167 + 0.515i)2-s + (1.38 − 1.00i)4-s + (−1.16 − 1.91i)5-s + 1.08·7-s + (1.62 + 1.18i)8-s + (0.789 − 0.918i)10-s + (−0.900 − 2.77i)11-s + (0.298 − 0.918i)13-s + (0.182 + 0.560i)14-s + (0.718 − 2.21i)16-s + (2.15 + 1.56i)17-s + (1.59 + 1.15i)19-s + (−3.52 − 1.47i)20-s + (1.27 − 0.928i)22-s + (2.11 + 6.49i)23-s + ⋯ |
L(s) = 1 | + (0.118 + 0.364i)2-s + (0.690 − 0.501i)4-s + (−0.519 − 0.854i)5-s + 0.411·7-s + (0.574 + 0.417i)8-s + (0.249 − 0.290i)10-s + (−0.271 − 0.835i)11-s + (0.0828 − 0.254i)13-s + (0.0487 + 0.149i)14-s + (0.179 − 0.552i)16-s + (0.521 + 0.379i)17-s + (0.364 + 0.265i)19-s + (−0.787 − 0.328i)20-s + (0.272 − 0.197i)22-s + (0.440 + 1.35i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42149 - 0.298903i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42149 - 0.298903i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.16 + 1.91i)T \) |
good | 2 | \( 1 + (-0.167 - 0.515i)T + (-1.61 + 1.17i)T^{2} \) |
| 7 | \( 1 - 1.08T + 7T^{2} \) |
| 11 | \( 1 + (0.900 + 2.77i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.298 + 0.918i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.15 - 1.56i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.59 - 1.15i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.11 - 6.49i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (5.51 - 4.00i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.01 + 1.46i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.54 - 7.82i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.26 + 10.0i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 7.72T + 43T^{2} \) |
| 47 | \( 1 + (-7.29 + 5.30i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (4.47 - 3.25i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (4.08 - 12.5i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.12 - 3.47i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (7.15 + 5.20i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-6.31 + 4.59i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.82 - 14.8i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-5.34 + 3.88i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-1.53 - 1.11i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (0.353 + 1.08i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-5.01 + 3.64i)T + (29.9 - 92.2i)T^{2} \) |
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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
−12.02240250871811857872293165694, −11.30596421121484131982837167419, −10.41264652228567589886441481085, −9.112754104239991095829638796436, −8.032558969890887991892808250637, −7.29220672406547107329622142953, −5.75607604364043995620845180576, −5.14973037234227306340817909997, −3.49331128554454642804183873245, −1.42983693016397954418139490119,
2.20566925660868365729496243891, 3.40220455279282279370913031340, 4.67211547795430209255162870956, 6.43779734663103922664176578432, 7.35854335272976701577834570795, 8.026986817265425012875704853493, 9.609532887365882556795889667092, 10.72474777395921969028776089990, 11.30311764173593554880877731134, 12.17531882746266400906242154312