L(s) = 1 | + (−10.3 − 4.47i)2-s + (−31.1 − 34.8i)3-s + (88 + 92.9i)4-s + 205. i·5-s + (168. + 501. i)6-s + 999. i·7-s + (−498. − 1.35e3i)8-s + (−242. + 2.17e3i)9-s + (920 − 2.13e3i)10-s + 1.97e3·11-s + (496. − 5.96e3i)12-s − 1.27e4·13-s + (4.46e3 − 1.03e4i)14-s + (7.17e3 − 6.41e3i)15-s + (−896. + 1.63e4i)16-s + 1.88e4i·17-s + ⋯ |
L(s) = 1 | + (−0.918 − 0.395i)2-s + (−0.666 − 0.745i)3-s + (0.687 + 0.726i)4-s + 0.735i·5-s + (0.317 + 0.948i)6-s + 1.10i·7-s + (−0.344 − 0.938i)8-s + (−0.111 + 0.993i)9-s + (0.290 − 0.676i)10-s + 0.447·11-s + (0.0829 − 0.996i)12-s − 1.60·13-s + (0.435 − 1.01i)14-s + (0.548 − 0.490i)15-s + (−0.0546 + 0.998i)16-s + 0.930i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0829 - 0.996i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0829 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.347197 + 0.319504i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.347197 + 0.319504i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (10.3 + 4.47i)T \) |
| 3 | \( 1 + (31.1 + 34.8i)T \) |
good | 5 | \( 1 - 205. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 999. iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 1.97e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.27e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.88e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 2.39e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 7.23e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.06e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 9.03e4iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 4.33e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.85e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + 7.08e4iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 5.63e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.48e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 7.58e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.14e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 8.11e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 2.65e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.59e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.23e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 1.01e7T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.88e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 7.24e6T + 8.07e13T^{2} \) |
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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
−18.78635050264062439493460936320, −17.82281417238184685075871422963, −16.66067526651286276239280385004, −14.86222314403663166850411516370, −12.50541146134483754784196071845, −11.61152712056322090738830172542, −9.996807994387519671235422656562, −7.938236002908821722675550354989, −6.31400819264188877647603214996, −2.22195042962927585475181372723,
0.46445618493903922016043319982, 4.92190864043536808766193358771, 7.09175021848274678470365949358, 9.232252792017005568639828688244, 10.40744884795875416383243501229, 11.96906524196204569769380771503, 14.41649818967935099540945785497, 16.03246256813220137257544441649, 16.89480368139900730956985224805, 17.72329543583695631847866608590