| L(s) = 1 | + (−0.697 + 0.960i)2-s + (−145. + 448. i)3-s + (1.26e3 + 3.89e3i)4-s + (1.88e4 − 1.36e4i)5-s + (−329. − 452. i)6-s + (−1.69e5 + 5.50e4i)7-s + (−9.24e3 − 3.00e3i)8-s + (2.49e5 + 1.81e5i)9-s + 2.76e4i·10-s + (−5.55e5 + 1.68e6i)11-s − 1.93e6·12-s + (−2.13e6 + 2.93e6i)13-s + (6.53e4 − 2.01e5i)14-s + (3.38e6 + 1.04e7i)15-s + (−1.35e7 + 9.85e6i)16-s + (2.26e6 + 3.11e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.0109 + 0.0150i)2-s + (−0.199 + 0.615i)3-s + (0.308 + 0.950i)4-s + (1.20 − 0.874i)5-s + (−0.00705 − 0.00970i)6-s + (−1.44 + 0.468i)7-s + (−0.0352 − 0.0114i)8-s + (0.470 + 0.341i)9-s + 0.0276i·10-s + (−0.313 + 0.949i)11-s − 0.646·12-s + (−0.441 + 0.607i)13-s + (0.00868 − 0.0267i)14-s + (0.297 + 0.915i)15-s + (−0.808 + 0.587i)16-s + (0.0938 + 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.750032 + 1.34555i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.750032 + 1.34555i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (5.55e5 - 1.68e6i)T \) |
| good | 2 | \( 1 + (0.697 - 0.960i)T + (-1.26e3 - 3.89e3i)T^{2} \) |
| 3 | \( 1 + (145. - 448. i)T + (-4.29e5 - 3.12e5i)T^{2} \) |
| 5 | \( 1 + (-1.88e4 + 1.36e4i)T + (7.54e7 - 2.32e8i)T^{2} \) |
| 7 | \( 1 + (1.69e5 - 5.50e4i)T + (1.11e10 - 8.13e9i)T^{2} \) |
| 13 | \( 1 + (2.13e6 - 2.93e6i)T + (-7.19e12 - 2.21e13i)T^{2} \) |
| 17 | \( 1 + (-2.26e6 - 3.11e6i)T + (-1.80e14 + 5.54e14i)T^{2} \) |
| 19 | \( 1 + (6.69e7 + 2.17e7i)T + (1.79e15 + 1.30e15i)T^{2} \) |
| 23 | \( 1 - 1.53e8T + 2.19e16T^{2} \) |
| 29 | \( 1 + (-6.56e8 + 2.13e8i)T + (2.86e17 - 2.07e17i)T^{2} \) |
| 31 | \( 1 + (-1.22e9 - 8.88e8i)T + (2.43e17 + 7.49e17i)T^{2} \) |
| 37 | \( 1 + (5.93e8 + 1.82e9i)T + (-5.32e18 + 3.86e18i)T^{2} \) |
| 41 | \( 1 + (-4.26e9 - 1.38e9i)T + (1.82e19 + 1.32e19i)T^{2} \) |
| 43 | \( 1 - 5.46e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (-3.96e9 + 1.22e10i)T + (-9.40e19 - 6.82e19i)T^{2} \) |
| 53 | \( 1 + (6.19e9 + 4.49e9i)T + (1.51e20 + 4.67e20i)T^{2} \) |
| 59 | \( 1 + (1.14e8 + 3.52e8i)T + (-1.43e21 + 1.04e21i)T^{2} \) |
| 61 | \( 1 + (-3.02e10 - 4.17e10i)T + (-8.20e20 + 2.52e21i)T^{2} \) |
| 67 | \( 1 - 7.40e9T + 8.18e21T^{2} \) |
| 71 | \( 1 + (-8.45e10 + 6.14e10i)T + (5.07e21 - 1.56e22i)T^{2} \) |
| 73 | \( 1 + (-7.29e10 + 2.36e10i)T + (1.85e22 - 1.34e22i)T^{2} \) |
| 79 | \( 1 + (1.02e10 - 1.41e10i)T + (-1.82e22 - 5.61e22i)T^{2} \) |
| 83 | \( 1 + (1.63e11 + 2.25e11i)T + (-3.30e22 + 1.01e23i)T^{2} \) |
| 89 | \( 1 + 1.07e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + (-5.91e11 - 4.29e11i)T + (2.14e23 + 6.59e23i)T^{2} \) |
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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
−17.49109951414927450033868918638, −16.60351381619909771518590373919, −15.61407419292690327743979193328, −13.15718845084980109584140698363, −12.50036610591393210205751276339, −10.12664368598288172318414538449, −9.019376608424447098857980715755, −6.63978323779219804202426336108, −4.64480734712972691828239391024, −2.38144195111618269968999708283,
0.72468959738178747314051707621, 2.66388131206825583910075423016, 6.06515139165986399366629437000, 6.71530314969649892426769594834, 9.789272404832876276007065799823, 10.59783509488035788915418545256, 12.89668839868875534770379022539, 13.97731575996221945310921322613, 15.50436334134421594212898719884, 17.21815364514696049427166020801
