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
−17.21815364514696049427166020801, −15.50436334134421594212898719884, −13.97731575996221945310921322613, −12.89668839868875534770379022539, −10.59783509488035788915418545256, −9.789272404832876276007065799823, −6.71530314969649892426769594834, −6.06515139165986399366629437000, −2.66388131206825583910075423016, −0.72468959738178747314051707621,
2.38144195111618269968999708283, 4.64480734712972691828239391024, 6.63978323779219804202426336108, 9.019376608424447098857980715755, 10.12664368598288172318414538449, 12.50036610591393210205751276339, 13.15718845084980109584140698363, 15.61407419292690327743979193328, 16.60351381619909771518590373919, 17.49109951414927450033868918638