| L(s) = 1 | + (−0.266 + 1.38i)2-s + (0.974 + 1.08i)3-s + (−1.85 − 0.738i)4-s + (1.14 + 1.98i)5-s + (−1.76 + 1.06i)6-s + (−2.24 + 0.235i)7-s + (1.52 − 2.38i)8-s + (0.0917 − 0.873i)9-s + (−3.06 + 1.06i)10-s + (2.10 + 0.936i)11-s + (−1.01 − 2.73i)12-s + (0.443 + 2.08i)13-s + (0.268 − 3.17i)14-s + (−1.03 + 3.18i)15-s + (2.90 + 2.74i)16-s + (−2.83 − 6.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.188 + 0.982i)2-s + (0.562 + 0.624i)3-s + (−0.929 − 0.369i)4-s + (0.513 + 0.889i)5-s + (−0.719 + 0.435i)6-s + (−0.846 + 0.0890i)7-s + (0.537 − 0.843i)8-s + (0.0305 − 0.291i)9-s + (−0.970 + 0.337i)10-s + (0.634 + 0.282i)11-s + (−0.291 − 0.788i)12-s + (0.123 + 0.579i)13-s + (0.0718 − 0.848i)14-s + (−0.266 + 0.821i)15-s + (0.726 + 0.686i)16-s + (−0.687 − 1.54i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.574598 + 0.910757i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.574598 + 0.910757i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.266 - 1.38i)T \) |
| 31 | \( 1 + (-5.07 + 2.29i)T \) |
| good | 3 | \( 1 + (-0.974 - 1.08i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + (-1.14 - 1.98i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2.24 - 0.235i)T + (6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (-2.10 - 0.936i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-0.443 - 2.08i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (2.83 + 6.36i)T + (-11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (0.255 - 1.20i)T + (-17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-2.62 - 1.90i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-4.35 + 1.41i)T + (23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-3.57 - 2.06i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.915 - 1.01i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (10.1 + 2.16i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (10.4 + 3.38i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (11.2 + 1.18i)T + (51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-9.01 + 8.11i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 5.73iT - 61T^{2} \) |
| 67 | \( 1 + (8.91 - 5.14i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.98 - 0.523i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (2.39 - 5.36i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (10.3 - 4.61i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-3.29 + 3.66i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (2.66 + 3.66i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.12 + 3.72i)T + (29.9 - 92.2i)T^{2} \) |
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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
−14.07935322307418462535686187608, −13.20879145989135822740009712283, −11.61465040751153930534692727052, −9.886865217630833556770665320927, −9.678667150939035592175100520578, −8.577013216513845470358456514749, −6.84988056932351745576632364813, −6.42618310457948571967562318833, −4.60185279311170705452994727618, −3.14385709159522989584359263379,
1.49760853867785385333192279104, 3.12137780244280263958396385189, 4.76231737407297658011667729592, 6.44210700307884601409647669303, 8.206688332939293273011292629092, 8.818040872935784552922173531588, 9.916209723994429845296527201446, 10.95625177823464334092013141714, 12.42195116695372081576815101496, 13.06063481372899265800879629004
