| L(s) = 1 | + (−0.104 − 0.994i)2-s + (−0.669 + 0.743i)3-s + (−0.978 + 0.207i)4-s + (2.67 − 1.19i)5-s + (0.809 + 0.587i)6-s + (2.49 + 0.880i)7-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (−1.46 − 2.53i)10-s + (−2.53 + 2.13i)11-s + (0.5 − 0.866i)12-s + (0.928 − 0.674i)13-s + (0.614 − 2.57i)14-s + (−0.904 + 2.78i)15-s + (0.913 − 0.406i)16-s + (−0.159 + 1.51i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 − 0.703i)2-s + (−0.386 + 0.429i)3-s + (−0.489 + 0.103i)4-s + (1.19 − 0.532i)5-s + (0.330 + 0.239i)6-s + (0.943 + 0.332i)7-s + (0.109 + 0.336i)8-s + (−0.0348 − 0.331i)9-s + (−0.462 − 0.801i)10-s + (−0.764 + 0.645i)11-s + (0.144 − 0.249i)12-s + (0.257 − 0.187i)13-s + (0.164 − 0.687i)14-s + (−0.233 + 0.718i)15-s + (0.228 − 0.101i)16-s + (−0.0386 + 0.367i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45450 - 0.403208i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45450 - 0.403208i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 + 0.994i)T \) |
| 3 | \( 1 + (0.669 - 0.743i)T \) |
| 7 | \( 1 + (-2.49 - 0.880i)T \) |
| 11 | \( 1 + (2.53 - 2.13i)T \) |
| good | 5 | \( 1 + (-2.67 + 1.19i)T + (3.34 - 3.71i)T^{2} \) |
| 13 | \( 1 + (-0.928 + 0.674i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.159 - 1.51i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-6.45 - 1.37i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-0.745 + 1.29i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.65 + 5.08i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.86 - 1.27i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (2.53 + 2.80i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (3.65 + 11.2i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 4.19T + 43T^{2} \) |
| 47 | \( 1 + (-0.802 - 0.170i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-11.3 - 5.04i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (13.1 - 2.80i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (9.69 - 4.31i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-0.555 - 0.961i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.46 - 3.24i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (9.18 - 1.95i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-1.04 - 9.90i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (10.8 + 7.87i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.58 + 2.74i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.96 - 5.05i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.76747261406008040397259934330, −10.20278875474122417488225956728, −9.380613260089997192264546878134, −8.580957386407404087732977126214, −7.46830209646006821913684591652, −5.75113784972290050452837388479, −5.29387474399975083819011515601, −4.27878269050291256540904157398, −2.58730167324667471717338070314, −1.39683189542581887573844914870,
1.35230124299465646719569539105, 2.95144790657827509497301274559, 4.87548627800750337583106060537, 5.54905926064370374144165743787, 6.46963442226746894018162954629, 7.37150625741621258878066029038, 8.192550950184762642805411823894, 9.306654663615382702928233896342, 10.25982038324078797928714627729, 11.01195406678740686214400699987
