| L(s) = 1 | + (1.03 − 0.966i)2-s + (0.167 + 1.11i)3-s + (0.133 − 1.99i)4-s + (−2.65 − 1.04i)5-s + (1.24 + 0.985i)6-s + (2.56 + 0.648i)7-s + (−1.78 − 2.19i)8-s + (1.66 − 0.512i)9-s + (−3.74 + 1.48i)10-s + (1.29 − 4.20i)11-s + (2.23 − 0.185i)12-s + (1.42 − 0.324i)13-s + (3.27 − 1.80i)14-s + (0.711 − 3.11i)15-s + (−3.96 − 0.533i)16-s + (−0.726 + 0.495i)17-s + ⋯ |
| L(s) = 1 | + (0.730 − 0.683i)2-s + (0.0966 + 0.641i)3-s + (0.0668 − 0.997i)4-s + (−1.18 − 0.465i)5-s + (0.508 + 0.402i)6-s + (0.969 + 0.245i)7-s + (−0.632 − 0.774i)8-s + (0.553 − 0.170i)9-s + (−1.18 + 0.469i)10-s + (0.390 − 1.26i)11-s + (0.646 − 0.0535i)12-s + (0.394 − 0.0900i)13-s + (0.875 − 0.483i)14-s + (0.183 − 0.804i)15-s + (−0.991 − 0.133i)16-s + (−0.176 + 0.120i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.49838 - 1.18518i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49838 - 1.18518i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.03 + 0.966i)T \) |
| 7 | \( 1 + (-2.56 - 0.648i)T \) |
| good | 3 | \( 1 + (-0.167 - 1.11i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (2.65 + 1.04i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-1.29 + 4.20i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-1.42 + 0.324i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (0.726 - 0.495i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-4.43 + 2.55i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.97 + 3.39i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (2.61 - 5.42i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (3.42 - 5.93i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.38 + 0.253i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (-2.18 - 2.74i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-6.60 - 5.26i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (1.71 - 1.59i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-10.7 - 0.808i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (10.0 - 3.94i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (8.05 - 0.603i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (11.5 + 6.66i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.46 + 0.703i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-11.2 - 10.4i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (5.25 + 9.11i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.53 + 0.806i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-5.35 + 1.65i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 9.24T + 97T^{2} \) |
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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
−11.15332884788571136474251331542, −10.64783245788899035424353684893, −9.264670158673880119938472505817, −8.595778350323056446635481421798, −7.42939980735409556921576127964, −5.95549647354540822197925097280, −4.82621125546428224804245665391, −4.11623445487771269580044249448, −3.21279867119681462823726450838, −1.17013821124766011351970590706,
2.01283161023687826271148197292, 3.86877861977159131603135439981, 4.38581290947264230156508993502, 5.80504480614459467374272417490, 7.17419055896081224977931134496, 7.52819817527321293457912792034, 8.071790331811876388775731378205, 9.559040559136398833743294939222, 10.97889500451614521526855103622, 11.89180433008776178745937804552
