| L(s) = 1 | + (31.7 − 18.3i)2-s + (65.0 + 112. i)3-s + (417. − 722. i)4-s − 1.41e3i·5-s + (4.13e3 + 2.38e3i)6-s + (6.50e3 + 3.75e3i)7-s − 1.18e4i·8-s + (1.38e3 − 2.40e3i)9-s + (−2.60e4 − 4.50e4i)10-s + (−3.71e4 + 2.14e4i)11-s + 1.08e5·12-s + (−1.00e5 − 2.10e4i)13-s + 2.75e5·14-s + (1.59e5 − 9.21e4i)15-s + (−3.47e3 − 6.01e3i)16-s + (−2.67e5 + 4.63e5i)17-s + ⋯ |
| L(s) = 1 | + (1.40 − 0.810i)2-s + (0.463 + 0.802i)3-s + (0.814 − 1.41i)4-s − 1.01i·5-s + (1.30 + 0.751i)6-s + (1.02 + 0.590i)7-s − 1.02i·8-s + (0.0705 − 0.122i)9-s + (−0.822 − 1.42i)10-s + (−0.764 + 0.441i)11-s + 1.51·12-s + (−0.978 − 0.204i)13-s + 1.91·14-s + (0.814 − 0.470i)15-s + (−0.0132 − 0.0229i)16-s + (−0.777 + 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.52738 - 1.37292i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.52738 - 1.37292i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (1.00e5 + 2.10e4i)T \) |
| good | 2 | \( 1 + (-31.7 + 18.3i)T + (256 - 443. i)T^{2} \) |
| 3 | \( 1 + (-65.0 - 112. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + 1.41e3iT - 1.95e6T^{2} \) |
| 7 | \( 1 + (-6.50e3 - 3.75e3i)T + (2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (3.71e4 - 2.14e4i)T + (1.17e9 - 2.04e9i)T^{2} \) |
| 17 | \( 1 + (2.67e5 - 4.63e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (3.62e5 + 2.09e5i)T + (1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.47e5 - 2.55e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (8.10e5 + 1.40e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + 3.25e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 + (1.73e7 - 1.00e7i)T + (6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-2.98e7 + 1.72e7i)T + (1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-1.56e6 + 2.71e6i)T + (-2.51e14 - 4.35e14i)T^{2} \) |
| 47 | \( 1 + 5.82e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + 2.43e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (-1.40e8 - 8.09e7i)T + (4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (4.21e7 - 7.30e7i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (4.23e7 - 2.44e7i)T + (1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (2.87e7 + 1.65e7i)T + (2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + 2.22e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 - 1.28e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.32e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + (-5.06e8 + 2.92e8i)T + (1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (8.94e8 + 5.16e8i)T + (3.80e17 + 6.58e17i)T^{2} \) |
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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
−17.44296895297923404621444827764, −15.36316546446503783258356104732, −14.81193958579167265795561898971, −13.08388232536301038383201091948, −12.10241278166952661636503337705, −10.47203279502703026251932058059, −8.644288618099399098112673100658, −5.20264544616574041824251221733, −4.26227499782834824633665324755, −2.15122543136600175696083174013,
2.63263028186040645115975940662, 4.81389102714685375285711352454, 6.90057935616463042835917462995, 7.72197537833508371550077393113, 10.96523936435327421845760502993, 12.75111558363915598668991043421, 14.06541599285912086404244099890, 14.45910062377295646349955358236, 16.03497371408126738152287761486, 17.80862464104291171783907509784
