| L(s) = 1 | + (−1.11 + 0.869i)2-s + (0.264 − 1.75i)3-s + (0.486 − 1.93i)4-s + (3.38 − 1.32i)5-s + (1.22 + 2.18i)6-s + (−0.357 − 2.62i)7-s + (1.14 + 2.58i)8-s + (−0.132 − 0.0408i)9-s + (−2.61 + 4.42i)10-s + (0.946 + 3.06i)11-s + (−3.26 − 1.36i)12-s + (4.21 + 0.961i)13-s + (2.67 + 2.61i)14-s + (−1.43 − 6.28i)15-s + (−3.52 − 1.88i)16-s + (−3.19 − 2.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.788 + 0.615i)2-s + (0.152 − 1.01i)3-s + (0.243 − 0.969i)4-s + (1.51 − 0.594i)5-s + (0.501 + 0.891i)6-s + (−0.135 − 0.990i)7-s + (0.404 + 0.914i)8-s + (−0.0441 − 0.0136i)9-s + (−0.828 + 1.39i)10-s + (0.285 + 0.925i)11-s + (−0.943 − 0.393i)12-s + (1.16 + 0.266i)13-s + (0.716 + 0.698i)14-s + (−0.370 − 1.62i)15-s + (−0.881 − 0.472i)16-s + (−0.774 − 0.527i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16265 - 0.580071i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16265 - 0.580071i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.11 - 0.869i)T \) |
| 7 | \( 1 + (0.357 + 2.62i)T \) |
| good | 3 | \( 1 + (-0.264 + 1.75i)T + (-2.86 - 0.884i)T^{2} \) |
| 5 | \( 1 + (-3.38 + 1.32i)T + (3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (-0.946 - 3.06i)T + (-9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (-4.21 - 0.961i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (3.19 + 2.17i)T + (6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (2.19 + 1.26i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.54 - 4.46i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (-3.71 - 7.71i)T + (-18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-1.03 - 1.79i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (9.23 + 0.692i)T + (36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (-5.03 + 6.31i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-0.212 + 0.169i)T + (9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (-2.62 - 2.43i)T + (3.51 + 46.8i)T^{2} \) |
| 53 | \( 1 + (-2.67 + 0.200i)T + (52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (1.20 + 0.473i)T + (43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (3.05 + 0.229i)T + (60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (4.44 - 2.56i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.15 + 1.51i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (4.56 - 4.23i)T + (5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (5.58 - 9.67i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.47 + 1.93i)T + (74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (3.08 + 0.951i)T + (73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 + 7.26T + 97T^{2} \) |
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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
−10.78760101446061851171633298465, −10.08881266049762815663264586687, −9.219580400492231772263004272747, −8.460521457445985593557952747057, −7.18836137226760085313644623674, −6.72799576892125319382341162540, −5.79297924568849295125719519163, −4.50811223609099962409392311979, −1.98645950248718964027269074997, −1.28671746280573515387971800835,
1.92749482016835228895090693114, 3.01871102816313720293789664521, 4.18992939527612481095696764298, 5.98738613292894869898230126713, 6.38403232656855493398364102867, 8.370332759500208645052561533106, 8.881082753144181492978816564134, 9.762337198639859670765433147075, 10.39518599675290319289017232700, 10.97185025997126939023454161503
