| L(s) = 1 | + (−0.913 − 0.406i)2-s + (0.578 − 0.420i)3-s + (0.669 + 0.743i)4-s + (1.23 − 0.262i)5-s + (−0.699 + 0.148i)6-s + (0.200 − 1.91i)7-s + (−0.309 − 0.951i)8-s + (−0.768 + 2.36i)9-s + (−1.23 − 0.262i)10-s + 5.96·11-s + (0.699 + 0.148i)12-s + (−3.04 − 5.26i)13-s + (−0.961 + 1.66i)14-s + (0.603 − 0.670i)15-s + (−0.104 + 0.994i)16-s + (−0.396 − 0.440i)17-s + ⋯ |
| L(s) = 1 | + (−0.645 − 0.287i)2-s + (0.334 − 0.242i)3-s + (0.334 + 0.371i)4-s + (0.551 − 0.117i)5-s + (−0.285 + 0.0607i)6-s + (0.0759 − 0.722i)7-s + (−0.109 − 0.336i)8-s + (−0.256 + 0.788i)9-s + (−0.390 − 0.0829i)10-s + 1.79·11-s + (0.202 + 0.0429i)12-s + (−0.843 − 1.46i)13-s + (−0.256 + 0.445i)14-s + (0.155 − 0.173i)15-s + (−0.0261 + 0.248i)16-s + (−0.0962 − 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.907737 - 0.320537i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.907737 - 0.320537i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.913 + 0.406i)T \) |
| 61 | \( 1 + (1.54 + 7.65i)T \) |
| good | 3 | \( 1 + (-0.578 + 0.420i)T + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (-1.23 + 0.262i)T + (4.56 - 2.03i)T^{2} \) |
| 7 | \( 1 + (-0.200 + 1.91i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 - 5.96T + 11T^{2} \) |
| 13 | \( 1 + (3.04 + 5.26i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.396 + 0.440i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.0598 - 0.569i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (2.81 - 8.67i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (2.34 - 4.06i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.883 - 0.393i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (3.43 + 2.49i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (2.23 + 1.62i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (5.68 - 6.31i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (-4.57 + 7.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.833 - 2.56i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (1.97 + 0.877i)T + (39.4 + 43.8i)T^{2} \) |
| 67 | \( 1 + (-7.73 + 1.64i)T + (61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (-2.11 - 0.448i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (5.07 + 1.07i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-4.24 + 4.71i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-7.99 - 3.56i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (13.2 - 9.61i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (4.39 - 1.95i)T + (64.9 - 72.0i)T^{2} \) |
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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
−13.41804670360978027970526625964, −12.23291537908577803228287522560, −11.14588307673054116505409489686, −10.06623896755235132340264709120, −9.230074303578976581510601987579, −7.958182867757217354952881865946, −7.08757506379226608219504666328, −5.47265189931491859717009026866, −3.54684508987348247976317349205, −1.69315530124456443111099463862,
2.15160202274095963368918147438, 4.20356923051290088272249127827, 6.11998012789624503926224138628, 6.80701133584292204743402608580, 8.633925110030600819858213589501, 9.222457612952919796725561094773, 9.989017060746206915525747649338, 11.67032432461418391300115472459, 12.13493806703341520328327031275, 14.08332375693461843870552964220
