| L(s) = 1 | + (−0.637 − 1.26i)2-s + (−0.420 + 2.78i)3-s + (−1.18 + 1.61i)4-s + (3.23 − 1.26i)5-s + (3.78 − 1.24i)6-s + (0.157 − 2.64i)7-s + (2.78 + 0.470i)8-s + (−4.73 − 1.46i)9-s + (−3.66 − 3.27i)10-s + (−1.48 − 4.80i)11-s + (−3.99 − 3.98i)12-s + (5.09 + 1.16i)13-s + (−3.43 + 1.48i)14-s + (2.17 + 9.54i)15-s + (−1.18 − 3.82i)16-s + (1.52 + 1.04i)17-s + ⋯ |
| L(s) = 1 | + (−0.450 − 0.892i)2-s + (−0.242 + 1.61i)3-s + (−0.593 + 0.805i)4-s + (1.44 − 0.567i)5-s + (1.54 − 0.509i)6-s + (0.0596 − 0.998i)7-s + (0.986 + 0.166i)8-s + (−1.57 − 0.487i)9-s + (−1.15 − 1.03i)10-s + (−0.447 − 1.45i)11-s + (−1.15 − 1.15i)12-s + (1.41 + 0.322i)13-s + (−0.917 + 0.396i)14-s + (0.562 + 2.46i)15-s + (−0.296 − 0.955i)16-s + (0.370 + 0.252i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22391 - 0.203444i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22391 - 0.203444i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.637 + 1.26i)T \) |
| 7 | \( 1 + (-0.157 + 2.64i)T \) |
| good | 3 | \( 1 + (0.420 - 2.78i)T + (-2.86 - 0.884i)T^{2} \) |
| 5 | \( 1 + (-3.23 + 1.26i)T + (3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (1.48 + 4.80i)T + (-9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (-5.09 - 1.16i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-1.52 - 1.04i)T + (6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (-3.89 - 2.24i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.05 - 2.08i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (1.89 + 3.94i)T + (-18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-2.30 - 3.98i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.27 - 0.395i)T + (36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (0.683 - 0.857i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (5.18 - 4.13i)T + (9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (3.45 + 3.20i)T + (3.51 + 46.8i)T^{2} \) |
| 53 | \( 1 + (-6.54 + 0.490i)T + (52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (-7.97 - 3.12i)T + (43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (5.57 + 0.417i)T + (60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (0.393 - 0.227i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.26 + 0.607i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (3.99 - 3.70i)T + (5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (-1.16 + 2.02i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.89 + 1.57i)T + (74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (15.3 + 4.74i)T + (73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 + 0.457T + 97T^{2} \) |
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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
−10.98075642394353398565034129838, −10.18055311892172583067770635165, −9.816493904691006869189264289322, −8.884121613280974271673963591373, −8.131445682864781932954724508303, −6.05198344575224040891602936058, −5.25730576511845199455232045686, −4.05117969316906988932389812594, −3.24819578894808395515882877094, −1.23253180839846200953514054248,
1.49115237060517063458081238608, 2.44571314627601821565649043371, 5.25217321408061530114460460607, 5.93420249767718965125807906229, 6.59707659813490938814637490943, 7.42872513602829478397546851316, 8.365395618234694148150186742624, 9.405898578963846980709299207835, 10.19494321760330962254934986137, 11.36988225913976117813752735670
