| L(s) = 1 | + (−0.326 − 1.37i)2-s + (2.89 + 0.797i)3-s + (−1.78 + 0.897i)4-s + (−7.40 + 5.51i)5-s + (0.154 − 4.23i)6-s + (−1.07 + 0.706i)7-s + (1.81 + 2.16i)8-s + (7.72 + 4.61i)9-s + (10.0 + 8.39i)10-s + (−1.07 + 9.21i)11-s + (−5.88 + 1.17i)12-s + (4.35 + 14.5i)13-s + (1.32 + 1.24i)14-s + (−25.8 + 10.0i)15-s + (2.38 − 3.20i)16-s + (0.471 + 0.0830i)17-s + ⋯ |
| L(s) = 1 | + (−0.163 − 0.688i)2-s + (0.964 + 0.265i)3-s + (−0.446 + 0.224i)4-s + (−1.48 + 1.10i)5-s + (0.0256 − 0.706i)6-s + (−0.153 + 0.100i)7-s + (0.227 + 0.270i)8-s + (0.858 + 0.512i)9-s + (1.00 + 0.839i)10-s + (−0.0978 + 0.837i)11-s + (−0.490 + 0.0975i)12-s + (0.335 + 1.11i)13-s + (0.0944 + 0.0890i)14-s + (−1.72 + 0.669i)15-s + (0.149 − 0.200i)16-s + (0.0277 + 0.00488i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.969867 + 0.672281i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.969867 + 0.672281i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.326 + 1.37i)T \) |
| 3 | \( 1 + (-2.89 - 0.797i)T \) |
| good | 5 | \( 1 + (7.40 - 5.51i)T + (7.17 - 23.9i)T^{2} \) |
| 7 | \( 1 + (1.07 - 0.706i)T + (19.4 - 44.9i)T^{2} \) |
| 11 | \( 1 + (1.07 - 9.21i)T + (-117. - 27.9i)T^{2} \) |
| 13 | \( 1 + (-4.35 - 14.5i)T + (-141. + 92.8i)T^{2} \) |
| 17 | \( 1 + (-0.471 - 0.0830i)T + (271. + 98.8i)T^{2} \) |
| 19 | \( 1 + (3.61 + 20.4i)T + (-339. + 123. i)T^{2} \) |
| 23 | \( 1 + (15.9 - 24.2i)T + (-209. - 485. i)T^{2} \) |
| 29 | \( 1 + (7.55 - 7.13i)T + (48.8 - 839. i)T^{2} \) |
| 31 | \( 1 + (0.914 - 15.7i)T + (-954. - 111. i)T^{2} \) |
| 37 | \( 1 + (-61.2 + 22.2i)T + (1.04e3 - 879. i)T^{2} \) |
| 41 | \( 1 + (-8.54 + 36.0i)T + (-1.50e3 - 754. i)T^{2} \) |
| 43 | \( 1 + (12.7 + 29.5i)T + (-1.26e3 + 1.34e3i)T^{2} \) |
| 47 | \( 1 + (-10.0 + 0.587i)T + (2.19e3 - 256. i)T^{2} \) |
| 53 | \( 1 + (-14.3 - 8.31i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (8.16 + 69.8i)T + (-3.38e3 + 802. i)T^{2} \) |
| 61 | \( 1 + (-67.7 - 34.0i)T + (2.22e3 + 2.98e3i)T^{2} \) |
| 67 | \( 1 + (-27.1 + 28.7i)T + (-261. - 4.48e3i)T^{2} \) |
| 71 | \( 1 + (57.4 - 68.4i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (97.8 - 82.0i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-115. + 27.4i)T + (5.57e3 - 2.80e3i)T^{2} \) |
| 83 | \( 1 + (-9.97 - 42.1i)T + (-6.15e3 + 3.09e3i)T^{2} \) |
| 89 | \( 1 + (-52.4 - 62.4i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (83.7 - 112. i)T + (-2.69e3 - 9.01e3i)T^{2} \) |
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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
−12.69357006177880304325131059347, −11.62764106971591020106353616794, −10.91933057990757157316116384380, −9.815535483633695098696682512153, −8.828131657202864969496861010982, −7.67395885244709723297588064037, −6.96122167524126859209504478296, −4.40002743669009443604035318668, −3.61466441935740750850038418714, −2.36364238396730523144159233917,
0.72505745512536035251904863425, 3.44662131358555911527514614581, 4.47551451945030485596149528721, 6.09578279507839755100140966592, 7.78309224160166186370286599203, 8.071961078403870716946168016675, 8.858851269942679176025961559011, 10.15580408205655329872552721988, 11.65186528158917938888543156072, 12.76955379658829394682656726366
