L(s) = 1 | + (23.9 − 21.2i)2-s − 94.4·3-s + (120. − 1.01e3i)4-s + (3.10e3 + 328. i)5-s + (−2.25e3 + 2.00e3i)6-s + 2.00e4·7-s + (−1.87e4 − 2.68e4i)8-s − 5.01e4·9-s + (8.13e4 − 5.81e4i)10-s − 1.98e5i·11-s + (−1.13e4 + 9.60e4i)12-s − 3.52e5i·13-s + (4.80e5 − 4.26e5i)14-s + (−2.93e5 − 3.10e4i)15-s + (−1.01e6 − 2.45e5i)16-s + 1.97e6i·17-s + ⋯ |
L(s) = 1 | + (0.747 − 0.664i)2-s − 0.388·3-s + (0.117 − 0.993i)4-s + (0.994 + 0.105i)5-s + (−0.290 + 0.258i)6-s + 1.19·7-s + (−0.571 − 0.820i)8-s − 0.848·9-s + (0.813 − 0.581i)10-s − 1.23i·11-s + (−0.0457 + 0.385i)12-s − 0.950i·13-s + (0.893 − 0.793i)14-s + (−0.386 − 0.0408i)15-s + (−0.972 − 0.233i)16-s + 1.39i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(1.72722 - 2.16370i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.72722 - 2.16370i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-23.9 + 21.2i)T \) |
| 5 | \( 1 + (-3.10e3 - 328. i)T \) |
good | 3 | \( 1 + 94.4T + 5.90e4T^{2} \) |
| 7 | \( 1 - 2.00e4T + 2.82e8T^{2} \) |
| 11 | \( 1 + 1.98e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 3.52e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 - 1.97e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 + 3.78e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 4.41e6T + 4.14e13T^{2} \) |
| 29 | \( 1 + 1.04e7T + 4.20e14T^{2} \) |
| 31 | \( 1 - 1.45e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 - 1.27e8iT - 4.80e15T^{2} \) |
| 41 | \( 1 - 9.37e7T + 1.34e16T^{2} \) |
| 43 | \( 1 - 2.01e8T + 2.16e16T^{2} \) |
| 47 | \( 1 + 6.61e7T + 5.25e16T^{2} \) |
| 53 | \( 1 + 8.60e7iT - 1.74e17T^{2} \) |
| 59 | \( 1 - 5.09e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 2.18e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + 7.20e8T + 1.82e18T^{2} \) |
| 71 | \( 1 - 1.16e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 1.07e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 - 3.63e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + 1.43e9T + 1.55e19T^{2} \) |
| 89 | \( 1 + 9.94e8T + 3.11e19T^{2} \) |
| 97 | \( 1 - 4.59e9iT - 7.37e19T^{2} \) |
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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
−15.09974847887935808933020991211, −14.07727457708040787273129939874, −13.01490573103766556777808938457, −11.30002094893302927523450094337, −10.66766075591153964781436144086, −8.709692116088383866261147329330, −6.07914705401625393732948978917, −5.10435902754896253340027794582, −2.82365944452140295609642467384, −1.06366901382129697706345143752,
2.12125583594077888644820932610, 4.67135395596031426897347852875, 5.78241894842360730033864771964, 7.42340721555109043771348693278, 9.170498601488965964843324587564, 11.25916006087973005674368429755, 12.43581908175137697427584912213, 14.10139159290014950789464365109, 14.58189852454764007865353504584, 16.42349307558995081285979695231