L(s) = 1 | + (−13.0 − 29.2i)2-s − 448. i·3-s + (−685. + 760. i)4-s − 1.39e3·5-s + (−1.31e4 + 5.84e3i)6-s + 1.93e4i·7-s + (3.11e4 + 1.01e4i)8-s − 1.42e5·9-s + (1.81e4 + 4.08e4i)10-s − 2.07e5i·11-s + (3.41e5 + 3.07e5i)12-s + 6.51e4·13-s + (5.65e5 − 2.51e5i)14-s + 6.27e5i·15-s + (−1.09e5 − 1.04e6i)16-s − 8.05e5·17-s + ⋯ |
L(s) = 1 | + (−0.406 − 0.913i)2-s − 1.84i·3-s + (−0.669 + 0.743i)4-s − 0.447·5-s + (−1.68 + 0.751i)6-s + 1.15i·7-s + (0.951 + 0.309i)8-s − 2.40·9-s + (0.181 + 0.408i)10-s − 1.28i·11-s + (1.37 + 1.23i)12-s + 0.175·13-s + (1.05 − 0.467i)14-s + 0.825i·15-s + (−0.104 − 0.994i)16-s − 0.567·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.743i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.669 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(0.0189898 + 0.00845473i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0189898 + 0.00845473i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (13.0 + 29.2i)T \) |
| 5 | \( 1 + 1.39e3T \) |
good | 3 | \( 1 + 448. iT - 5.90e4T^{2} \) |
| 7 | \( 1 - 1.93e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 2.07e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 6.51e4T + 1.37e11T^{2} \) |
| 17 | \( 1 + 8.05e5T + 2.01e12T^{2} \) |
| 19 | \( 1 - 4.08e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + 4.40e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 + 1.06e7T + 4.20e14T^{2} \) |
| 31 | \( 1 + 2.90e6iT - 8.19e14T^{2} \) |
| 37 | \( 1 + 1.45e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + 6.38e7T + 1.34e16T^{2} \) |
| 43 | \( 1 - 2.40e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 - 1.51e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 3.71e8T + 1.74e17T^{2} \) |
| 59 | \( 1 + 4.69e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 7.97e8T + 7.13e17T^{2} \) |
| 67 | \( 1 - 8.39e8iT - 1.82e18T^{2} \) |
| 71 | \( 1 + 4.87e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 3.05e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + 4.58e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 2.95e8iT - 1.55e19T^{2} \) |
| 89 | \( 1 + 3.76e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + 1.37e10T + 7.37e19T^{2} \) |
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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
−14.26041285822297613115214468845, −13.00040161753751471368097727884, −12.12373072044653292178192930060, −11.22195224747618731683732798556, −8.735575800626071798387480855543, −7.936685535672601435658896877204, −6.03284008391549787241601638871, −2.96298089212258536703315888481, −1.54788947822112263941658400160, −0.01044341629908163003304503329,
4.02316015453618254405685050975, 4.94965499225494605669986021786, 7.16687777061367211394987889980, 8.944125781714116022391344826971, 10.03481059399219208611067964833, 11.03551500338788280648649044494, 13.71740374432449866365668181785, 15.11950530605686762710706087615, 15.61638784863825840563200537022, 16.84810624676600089604346580008