L(s) = 1 | + 45.2i·2-s + (4.41 + 728. i)3-s − 2.04e3·4-s − 1.79e3i·5-s + (−3.29e4 + 199. i)6-s − 1.36e5·7-s − 9.26e4i·8-s + (−5.31e5 + 6.43e3i)9-s + 8.11e4·10-s + 1.76e6i·11-s + (−9.03e3 − 1.49e6i)12-s + 6.95e6·13-s − 6.18e6i·14-s + (1.30e6 − 7.91e3i)15-s + 4.19e6·16-s + 3.90e7i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.00605 + 0.999i)3-s − 0.500·4-s − 0.114i·5-s + (−0.707 + 0.00427i)6-s − 1.16·7-s − 0.353i·8-s + (−0.999 + 0.0121i)9-s + 0.0811·10-s + 0.993i·11-s + (−0.00302 − 0.499i)12-s + 1.44·13-s − 0.821i·14-s + (0.114 − 0.000694i)15-s + 0.250·16-s + 1.61i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00605i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.999 + 0.00605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(0.00300231 - 0.992271i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00300231 - 0.992271i\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 45.2iT \) |
| 3 | \( 1 + (-4.41 - 728. i)T \) |
good | 5 | \( 1 + 1.79e3iT - 2.44e8T^{2} \) |
| 7 | \( 1 + 1.36e5T + 1.38e10T^{2} \) |
| 11 | \( 1 - 1.76e6iT - 3.13e12T^{2} \) |
| 13 | \( 1 - 6.95e6T + 2.32e13T^{2} \) |
| 17 | \( 1 - 3.90e7iT - 5.82e14T^{2} \) |
| 19 | \( 1 + 6.02e7T + 2.21e15T^{2} \) |
| 23 | \( 1 - 1.16e8iT - 2.19e16T^{2} \) |
| 29 | \( 1 + 1.81e8iT - 3.53e17T^{2} \) |
| 31 | \( 1 - 2.39e8T + 7.87e17T^{2} \) |
| 37 | \( 1 + 1.10e8T + 6.58e18T^{2} \) |
| 41 | \( 1 + 3.21e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 - 6.08e9T + 3.99e19T^{2} \) |
| 47 | \( 1 - 3.06e9iT - 1.16e20T^{2} \) |
| 53 | \( 1 + 2.76e10iT - 4.91e20T^{2} \) |
| 59 | \( 1 - 7.62e10iT - 1.77e21T^{2} \) |
| 61 | \( 1 - 3.69e10T + 2.65e21T^{2} \) |
| 67 | \( 1 + 5.88e10T + 8.18e21T^{2} \) |
| 71 | \( 1 - 1.03e11iT - 1.64e22T^{2} \) |
| 73 | \( 1 + 1.32e11T + 2.29e22T^{2} \) |
| 79 | \( 1 - 1.57e11T + 5.90e22T^{2} \) |
| 83 | \( 1 + 3.02e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 + 5.40e10iT - 2.46e23T^{2} \) |
| 97 | \( 1 + 1.33e11T + 6.93e23T^{2} \) |
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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
−21.03589885794088482287524330783, −19.37307585631186529512361477512, −17.32440871203312106465694628707, −16.04318673393528944742308922373, −14.97545168242198303995359744912, −12.98220827451204007180900712829, −10.37058294287042598683520438129, −8.787728935165934326639790483133, −6.16691504109089348090783432642, −3.94161750708168357409220545920,
0.60671937747199961067823339247, 2.98783260521782573627115847554, 6.37300899312216697516961753252, 8.774278129596321957654829837643, 11.05463723265938482434783667408, 12.75180564268946181586688757785, 13.84819788717521938121398068858, 16.40246511746883477196878208251, 18.38068027590648892330590943407, 19.15130035503680744109969518438