| L(s) = 1 | + (0.145 − 1.40i)2-s + (−0.0665 − 0.0834i)3-s + (−1.95 − 0.408i)4-s + (1.11 − 0.890i)5-s + (−0.127 + 0.0814i)6-s + (−1.34 − 2.27i)7-s + (−0.859 + 2.69i)8-s + (0.665 − 2.91i)9-s + (−1.09 − 1.70i)10-s + (−1.57 + 0.359i)11-s + (0.0961 + 0.190i)12-s + (0.776 − 0.177i)13-s + (−3.40 + 1.55i)14-s + (−0.148 − 0.0339i)15-s + (3.66 + 1.60i)16-s + (1.17 − 2.44i)17-s + ⋯ |
| L(s) = 1 | + (0.102 − 0.994i)2-s + (−0.0384 − 0.0481i)3-s + (−0.978 − 0.204i)4-s + (0.499 − 0.398i)5-s + (−0.0518 + 0.0332i)6-s + (−0.507 − 0.861i)7-s + (−0.303 + 0.952i)8-s + (0.221 − 0.971i)9-s + (−0.344 − 0.537i)10-s + (−0.474 + 0.108i)11-s + (0.0277 + 0.0549i)12-s + (0.215 − 0.0491i)13-s + (−0.909 + 0.416i)14-s + (−0.0383 − 0.00875i)15-s + (0.916 + 0.400i)16-s + (0.285 − 0.593i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.676 + 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.442063 - 1.00639i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.442063 - 1.00639i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.145 + 1.40i)T \) |
| 7 | \( 1 + (1.34 + 2.27i)T \) |
| good | 3 | \( 1 + (0.0665 + 0.0834i)T + (-0.667 + 2.92i)T^{2} \) |
| 5 | \( 1 + (-1.11 + 0.890i)T + (1.11 - 4.87i)T^{2} \) |
| 11 | \( 1 + (1.57 - 0.359i)T + (9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (-0.776 + 0.177i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-1.17 + 2.44i)T + (-10.5 - 13.2i)T^{2} \) |
| 19 | \( 1 + 0.719T + 19T^{2} \) |
| 23 | \( 1 + (-1.19 - 2.49i)T + (-14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (-3.80 - 1.83i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 - 7.03T + 31T^{2} \) |
| 37 | \( 1 + (-4.28 - 2.06i)T + (23.0 + 28.9i)T^{2} \) |
| 41 | \( 1 + (0.875 - 0.697i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-4.71 - 3.75i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (2.47 + 10.8i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-2.38 + 1.14i)T + (33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 + (-8.49 + 10.6i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (4.31 - 8.95i)T + (-38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 - 2.88iT - 67T^{2} \) |
| 71 | \( 1 + (-5.12 - 10.6i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (12.1 + 2.76i)T + (65.7 + 31.6i)T^{2} \) |
| 79 | \( 1 - 9.74iT - 79T^{2} \) |
| 83 | \( 1 + (1.87 - 8.19i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (14.2 + 3.26i)T + (80.1 + 38.6i)T^{2} \) |
| 97 | \( 1 + 1.72iT - 97T^{2} \) |
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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
−12.19310196199863837583436697851, −11.20681652885477815320466327789, −10.00412580571617084446762336462, −9.615646575768828195631721442712, −8.400046244856297833161255978013, −6.91198650816422109500897980087, −5.55690848780836364230990286836, −4.26065452701599219306413351151, −3.03215516838233865046455311716, −1.04710076390748662971430303562,
2.69533899470714895755467944584, 4.51801021622617407363719815332, 5.74466320475959903705797983310, 6.45637380444187991729280989899, 7.80242406009617189118765897568, 8.655491683964041868713238895496, 9.831899126393925475244722937134, 10.61488924170481838937681369093, 12.19551965102831570196847467793, 13.09592876150736419379387827907
