| L(s) = 1 | + (1.40 − 0.173i)2-s + (−0.470 − 0.589i)3-s + (1.94 − 0.486i)4-s + (0.701 − 0.559i)5-s + (−0.762 − 0.746i)6-s + (−0.938 + 2.47i)7-s + (2.63 − 1.01i)8-s + (0.540 − 2.37i)9-s + (0.888 − 0.907i)10-s + (−0.406 + 0.0927i)11-s + (−1.19 − 0.915i)12-s + (0.0704 − 0.0160i)13-s + (−0.889 + 3.63i)14-s + (−0.660 − 0.150i)15-s + (3.52 − 1.88i)16-s + (−1.30 + 2.70i)17-s + ⋯ |
| L(s) = 1 | + (0.992 − 0.122i)2-s + (−0.271 − 0.340i)3-s + (0.970 − 0.243i)4-s + (0.313 − 0.250i)5-s + (−0.311 − 0.304i)6-s + (−0.354 + 0.934i)7-s + (0.932 − 0.359i)8-s + (0.180 − 0.790i)9-s + (0.280 − 0.286i)10-s + (−0.122 + 0.0279i)11-s + (−0.346 − 0.264i)12-s + (0.0195 − 0.00445i)13-s + (−0.237 + 0.971i)14-s + (−0.170 − 0.0389i)15-s + (0.881 − 0.471i)16-s + (−0.316 + 0.656i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.88417 - 0.493081i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88417 - 0.493081i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 + 0.173i)T \) |
| 7 | \( 1 + (0.938 - 2.47i)T \) |
| good | 3 | \( 1 + (0.470 + 0.589i)T + (-0.667 + 2.92i)T^{2} \) |
| 5 | \( 1 + (-0.701 + 0.559i)T + (1.11 - 4.87i)T^{2} \) |
| 11 | \( 1 + (0.406 - 0.0927i)T + (9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (-0.0704 + 0.0160i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (1.30 - 2.70i)T + (-10.5 - 13.2i)T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 23 | \( 1 + (-0.339 - 0.706i)T + (-14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (-4.62 - 2.22i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 - 2.54T + 31T^{2} \) |
| 37 | \( 1 + (7.13 + 3.43i)T + (23.0 + 28.9i)T^{2} \) |
| 41 | \( 1 + (5.37 - 4.28i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-3.75 - 2.99i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (0.549 + 2.40i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (5.53 - 2.66i)T + (33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 + (-6.25 + 7.84i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (-5.42 + 11.2i)T + (-38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 + (-3.56 - 7.40i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-11.0 - 2.52i)T + (65.7 + 31.6i)T^{2} \) |
| 79 | \( 1 + 8.93iT - 79T^{2} \) |
| 83 | \( 1 + (0.295 - 1.29i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-3.97 - 0.906i)T + (80.1 + 38.6i)T^{2} \) |
| 97 | \( 1 - 12.4iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.69908712889988580060794344899, −11.77180789922614645923417219020, −10.73530279993332676426557844172, −9.554714235403499093949160292713, −8.371188730744883722003111973377, −6.72059300216607524674509267186, −6.14963027578595144096509393403, −5.00817065810136705602475835270, −3.54081104856381181197745571369, −1.96789192094837810030991889595,
2.41076258058828814050132025864, 4.03717959705686027450843392956, 4.92601323255985730878188095979, 6.28133916809896860464113566023, 7.10801075186352128246875888143, 8.339931088187970050713209752455, 10.23974695503839588955374573177, 10.52894942277785424312119475991, 11.64335349726221773217343307181, 12.79509701625209941065141745762
