| L(s) = 1 | + (1.04 + 2.17i)2-s + (−3.26 − 4.78i)3-s + (36.2 − 45.4i)4-s + (−105. + 113. i)5-s + (6.98 − 12.0i)6-s + (260. − 150. i)7-s + (287. + 65.5i)8-s + (254. − 647. i)9-s + (−356. − 109. i)10-s + (−671. − 841. i)11-s + (−335. − 25.1i)12-s + (608. − 187. i)13-s + (599. + 408. i)14-s + (885. + 133. i)15-s + (−670. − 2.93e3i)16-s + (5.15e3 − 4.77e3i)17-s + ⋯ |
| L(s) = 1 | + (0.130 + 0.271i)2-s + (−0.120 − 0.177i)3-s + (0.566 − 0.710i)4-s + (−0.841 + 0.906i)5-s + (0.0323 − 0.0559i)6-s + (0.759 − 0.438i)7-s + (0.561 + 0.128i)8-s + (0.348 − 0.888i)9-s + (−0.356 − 0.109i)10-s + (−0.504 − 0.632i)11-s + (−0.194 − 0.0145i)12-s + (0.276 − 0.0853i)13-s + (0.218 + 0.148i)14-s + (0.262 + 0.0395i)15-s + (−0.163 − 0.717i)16-s + (1.04 − 0.972i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.71922 - 0.768839i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71922 - 0.768839i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (6.17e4 + 5.00e4i)T \) |
| good | 2 | \( 1 + (-1.04 - 2.17i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (3.26 + 4.78i)T + (-266. + 678. i)T^{2} \) |
| 5 | \( 1 + (105. - 113. i)T + (-1.16e3 - 1.55e4i)T^{2} \) |
| 7 | \( 1 + (-260. + 150. i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (671. + 841. i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (-608. + 187. i)T + (3.98e6 - 2.71e6i)T^{2} \) |
| 17 | \( 1 + (-5.15e3 + 4.77e3i)T + (1.80e6 - 2.40e7i)T^{2} \) |
| 19 | \( 1 + (-1.02e4 + 4.03e3i)T + (3.44e7 - 3.19e7i)T^{2} \) |
| 23 | \( 1 + (7.81e3 - 1.17e3i)T + (1.41e8 - 4.36e7i)T^{2} \) |
| 29 | \( 1 + (2.23e4 - 3.27e4i)T + (-2.17e8 - 5.53e8i)T^{2} \) |
| 31 | \( 1 + (-1.39e3 + 1.86e4i)T + (-8.77e8 - 1.32e8i)T^{2} \) |
| 37 | \( 1 + (-3.79e4 - 2.19e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (9.26e4 - 4.46e4i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (1.83e4 - 2.29e4i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (-2.13e5 - 6.59e4i)T + (1.83e10 + 1.24e10i)T^{2} \) |
| 59 | \( 1 + (2.82e4 + 1.23e5i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-1.95e5 + 1.46e4i)T + (5.09e10 - 7.67e9i)T^{2} \) |
| 67 | \( 1 + (-1.25e4 - 3.19e4i)T + (-6.63e10 + 6.15e10i)T^{2} \) |
| 71 | \( 1 + (-3.32e3 + 2.20e4i)T + (-1.22e11 - 3.77e10i)T^{2} \) |
| 73 | \( 1 + (-1.14e5 - 3.71e5i)T + (-1.25e11 + 8.52e10i)T^{2} \) |
| 79 | \( 1 + (-3.19e5 - 5.52e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-4.97e5 + 3.39e5i)T + (1.19e11 - 3.04e11i)T^{2} \) |
| 89 | \( 1 + (-1.85e5 - 2.72e5i)T + (-1.81e11 + 4.62e11i)T^{2} \) |
| 97 | \( 1 + (9.35e5 + 1.17e6i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−14.68880406349257351621756754998, −13.72903765187979133149917167602, −11.73115419376162971642640571182, −11.16831292584551703933900974394, −9.874639986405306294998606857825, −7.71884552595340226572359078254, −6.93881438041017507416564408652, −5.35735859569485001410374319464, −3.31564768646785140306405327472, −0.956794888369969962130995587849,
1.78174378655781671145636439104, 3.90478851766065905058710806817, 5.24435917216328162864001435091, 7.69901474821214359254563821518, 8.173485947299121591443492904339, 10.20419893273158713311218083745, 11.61345965693788040531803657257, 12.22547228911288681093541254376, 13.37490141637994878070715190924, 15.11252016076337037218678827340
