L(s) = 1 | + (20.2 + 40.4i)2-s + 712. i·3-s + (−1.22e3 + 1.64e3i)4-s − 7.12e3i·5-s + (−2.88e4 + 1.44e4i)6-s + 4.69e4·7-s + (−9.12e4 − 1.63e4i)8-s − 3.29e5·9-s + (2.88e5 − 1.44e5i)10-s + 6.77e5i·11-s + (−1.16e6 − 8.73e5i)12-s + 4.38e5i·13-s + (9.50e5 + 1.89e6i)14-s + 5.07e6·15-s + (−1.18e6 − 4.02e6i)16-s + 5.66e6·17-s + ⋯ |
L(s) = 1 | + (0.447 + 0.894i)2-s + 1.69i·3-s + (−0.598 + 0.800i)4-s − 1.01i·5-s + (−1.51 + 0.757i)6-s + 1.05·7-s + (−0.984 − 0.176i)8-s − 1.86·9-s + (0.911 − 0.456i)10-s + 1.26i·11-s + (−1.35 − 1.01i)12-s + 0.327i·13-s + (0.472 + 0.943i)14-s + 1.72·15-s + (−0.282 − 0.959i)16-s + 0.967·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.176i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.163999 + 1.83974i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.163999 + 1.83974i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-20.2 - 40.4i)T \) |
good | 3 | \( 1 - 712. iT - 1.77e5T^{2} \) |
| 5 | \( 1 + 7.12e3iT - 4.88e7T^{2} \) |
| 7 | \( 1 - 4.69e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 6.77e5iT - 2.85e11T^{2} \) |
| 13 | \( 1 - 4.38e5iT - 1.79e12T^{2} \) |
| 17 | \( 1 - 5.66e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 2.20e6iT - 1.16e14T^{2} \) |
| 23 | \( 1 + 2.68e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 4.65e6iT - 1.22e16T^{2} \) |
| 31 | \( 1 - 2.65e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 2.71e8iT - 1.77e17T^{2} \) |
| 41 | \( 1 - 6.32e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.17e9iT - 9.29e17T^{2} \) |
| 47 | \( 1 - 3.27e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 3.75e9iT - 9.26e18T^{2} \) |
| 59 | \( 1 + 1.31e9iT - 3.01e19T^{2} \) |
| 61 | \( 1 + 4.24e9iT - 4.35e19T^{2} \) |
| 67 | \( 1 - 7.41e9iT - 1.22e20T^{2} \) |
| 71 | \( 1 + 1.51e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 9.14e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.97e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.15e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 + 8.86e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.71e10T + 7.15e21T^{2} \) |
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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
−20.61678879269524471603793953606, −17.52348615761909959683825704365, −16.53160616953674555021147272079, −15.37976811647996632895276009187, −14.29584133500898568029402825953, −12.06537350611785443026099349064, −9.756718274268024600407852576151, −8.284391238342362881158530024142, −5.15245460957330976067858322575, −4.28282134250115599272602894963,
1.04170393479732175972278422105, 2.74515053180284408355831802381, 6.03535104036681206152845830956, 8.030496329685941066506949008374, 10.93347263178154298684516557149, 12.04368053873886821201844353137, 13.68179933591812552374930864963, 14.47857729584869633302861823399, 17.78590965707267749329759264352, 18.61238740359672518519817586468