L(s) = 1 | + (1.29 + 0.559i)2-s + (−1.60 + 0.653i)3-s + (1.37 + 1.45i)4-s + 3.97i·5-s + (−2.44 − 0.0487i)6-s + (0.969 + 2.65i)8-s + (2.14 − 2.09i)9-s + (−2.22 + 5.15i)10-s − 3.30·11-s + (−3.15 − 1.43i)12-s + 0.850·13-s + (−2.59 − 6.37i)15-s + (−0.228 + 3.99i)16-s − 0.576i·17-s + (3.95 − 1.52i)18-s − 3.59i·19-s + ⋯ |
L(s) = 1 | + (0.918 + 0.395i)2-s + (−0.926 + 0.377i)3-s + (0.686 + 0.726i)4-s + 1.77i·5-s + (−0.999 − 0.0199i)6-s + (0.342 + 0.939i)8-s + (0.715 − 0.699i)9-s + (−0.703 + 1.63i)10-s − 0.996·11-s + (−0.910 − 0.414i)12-s + 0.235·13-s + (−0.670 − 1.64i)15-s + (−0.0570 + 0.998i)16-s − 0.139i·17-s + (0.933 − 0.358i)18-s − 0.823i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.356569 + 1.64515i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.356569 + 1.64515i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.29 - 0.559i)T \) |
| 3 | \( 1 + (1.60 - 0.653i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3.97iT - 5T^{2} \) |
| 11 | \( 1 + 3.30T + 11T^{2} \) |
| 13 | \( 1 - 0.850T + 13T^{2} \) |
| 17 | \( 1 + 0.576iT - 17T^{2} \) |
| 19 | \( 1 + 3.59iT - 19T^{2} \) |
| 23 | \( 1 - 3.96T + 23T^{2} \) |
| 29 | \( 1 - 3.37iT - 29T^{2} \) |
| 31 | \( 1 + 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 4.29T + 37T^{2} \) |
| 41 | \( 1 - 5.35iT - 41T^{2} \) |
| 43 | \( 1 - 7.66iT - 43T^{2} \) |
| 47 | \( 1 + 9.09T + 47T^{2} \) |
| 53 | \( 1 + 0.608iT - 53T^{2} \) |
| 59 | \( 1 - 7.87T + 59T^{2} \) |
| 61 | \( 1 - 9.74T + 61T^{2} \) |
| 67 | \( 1 - 9.04iT - 67T^{2} \) |
| 71 | \( 1 - 9.15T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 - 4.13T + 83T^{2} \) |
| 89 | \( 1 - 5.75iT - 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−11.21572737762576940926000674158, −10.58463384468769554141932577021, −9.667125955500813746168344427164, −7.998935700056708782040634649162, −7.05864167273168176585280605354, −6.53819361816600406161532005489, −5.63641602957630996501009387688, −4.68068636855112214031284643478, −3.45614461259482342446070531523, −2.57935722072271030332345056854,
0.791298490640943696436118898623, 1.99393406976163929265279931342, 3.88694214304201082218352526760, 5.01313613333456824275696423044, 5.31205419418284317357655663760, 6.30501266063859822833151037582, 7.53739070542758517910289482848, 8.479236325245404527478317091632, 9.731236259680045955280197807984, 10.52604378630863070959046408521