| L(s) = 1 | + (−0.520 − 1.31i)2-s + (1.32 + 0.902i)3-s + (−1.45 + 1.36i)4-s + (0.323 + 0.0242i)5-s + (0.497 − 2.20i)6-s + (1.28 + 2.31i)7-s + (2.55 + 1.20i)8-s + (−0.159 − 0.405i)9-s + (−0.136 − 0.438i)10-s + (2.61 + 1.02i)11-s + (−3.16 + 0.496i)12-s + (2.01 − 1.60i)13-s + (2.37 − 2.89i)14-s + (0.406 + 0.323i)15-s + (0.251 − 3.99i)16-s + (−0.295 − 0.317i)17-s + ⋯ |
| L(s) = 1 | + (−0.368 − 0.929i)2-s + (0.763 + 0.520i)3-s + (−0.729 + 0.684i)4-s + (0.144 + 0.0108i)5-s + (0.203 − 0.901i)6-s + (0.484 + 0.874i)7-s + (0.904 + 0.425i)8-s + (−0.0530 − 0.135i)9-s + (−0.0431 − 0.138i)10-s + (0.787 + 0.309i)11-s + (−0.913 + 0.143i)12-s + (0.558 − 0.445i)13-s + (0.635 − 0.772i)14-s + (0.104 + 0.0836i)15-s + (0.0629 − 0.998i)16-s + (−0.0715 − 0.0771i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24791 - 0.173097i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24791 - 0.173097i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.520 + 1.31i)T \) |
| 7 | \( 1 + (-1.28 - 2.31i)T \) |
| good | 3 | \( 1 + (-1.32 - 0.902i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.323 - 0.0242i)T + (4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-2.61 - 1.02i)T + (8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-2.01 + 1.60i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (0.295 + 0.317i)T + (-1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (3.19 - 5.53i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.32 + 3.57i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (0.987 + 4.32i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (1.28 + 2.21i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (8.40 + 2.59i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-3.27 - 6.81i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (2.17 - 4.51i)T + (-26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (5.45 - 0.822i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-5.41 + 1.67i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.182 - 2.43i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-2.02 + 6.56i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-0.882 + 0.509i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (11.9 + 2.72i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-1.03 + 6.87i)T + (-69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (9.00 + 5.19i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.97 + 4.98i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-12.6 + 4.97i)T + (65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 - 8.05iT - 97T^{2} \) |
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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
−12.24813597481240102101667121353, −11.51955598169283651145911836156, −10.34933295083916743368988695586, −9.459826317910571813350786777435, −8.700440038653949371977818658917, −7.981387138414777640603171895211, −6.06180927799034833066071214647, −4.43820668687267455490471516523, −3.35994826789940976942315043096, −1.97206429904951648826285523748,
1.56338114816078794461764717030, 3.84430976729850263897040642134, 5.23186605284252143871694187635, 6.76062110219825691660503441894, 7.35396844672096316495025399900, 8.566976238368070669968957406058, 9.038024259879159491343031711113, 10.46550260505941380842995120044, 11.36528768435996488363943699397, 13.12542196275629208009953813292
