| L(s) = 1 | + (−1.33 − 0.454i)2-s + (−0.353 − 1.69i)3-s + (1.58 + 1.21i)4-s + (0.205 + 0.307i)5-s + (−0.297 + 2.43i)6-s + (1.38 − 3.34i)7-s + (−1.56 − 2.35i)8-s + (−2.74 + 1.19i)9-s + (−0.135 − 0.505i)10-s + (4.03 + 0.802i)11-s + (1.50 − 3.12i)12-s + (−1.78 − 1.19i)13-s + (−3.37 + 3.85i)14-s + (0.448 − 0.457i)15-s + (1.03 + 3.86i)16-s + (−4.80 − 4.80i)17-s + ⋯ |
| L(s) = 1 | + (−0.946 − 0.321i)2-s + (−0.204 − 0.978i)3-s + (0.793 + 0.609i)4-s + (0.0918 + 0.137i)5-s + (−0.121 + 0.992i)6-s + (0.524 − 1.26i)7-s + (−0.554 − 0.831i)8-s + (−0.916 + 0.399i)9-s + (−0.0427 − 0.159i)10-s + (1.21 + 0.242i)11-s + (0.434 − 0.900i)12-s + (−0.496 − 0.331i)13-s + (−0.903 + 1.02i)14-s + (0.115 − 0.118i)15-s + (0.257 + 0.966i)16-s + (−1.16 − 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.374435 - 0.619248i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.374435 - 0.619248i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.33 + 0.454i)T \) |
| 3 | \( 1 + (0.353 + 1.69i)T \) |
| good | 5 | \( 1 + (-0.205 - 0.307i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (-1.38 + 3.34i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-4.03 - 0.802i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (1.78 + 1.19i)T + (4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (4.80 + 4.80i)T + 17iT^{2} \) |
| 19 | \( 1 + (4.56 + 3.05i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-0.318 - 0.769i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-0.933 - 4.69i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 - 3.86T + 31T^{2} \) |
| 37 | \( 1 + (-5.93 - 8.88i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (-2.81 + 1.16i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-0.497 - 0.0989i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (-3.50 + 3.50i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.878 - 4.41i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (-7.88 + 5.27i)T + (22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (-1.65 - 8.31i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (-0.680 + 0.135i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (-7.27 - 3.01i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (6.01 - 2.49i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-6.21 + 6.21i)T - 79iT^{2} \) |
| 83 | \( 1 + (1.80 - 2.70i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (4.83 + 2.00i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 13.4iT - 97T^{2} \) |
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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
−11.95043124288485253717225702579, −11.23392668661821881172695869801, −10.40752677148676291632196120236, −9.130669181818725183294618139355, −8.131784129559208901179855310247, −6.98653552205821271295686919520, −6.65897316703272441768591909532, −4.45731225814501177124444593981, −2.51811013584577725786564030881, −0.925012568703953670837154865932,
2.19417516154698462220429246636, 4.27683573991432971608004914219, 5.71520738593201165714709869384, 6.45896748916897935097372695627, 8.297466512922599718009356272824, 8.917540076543592727790217635408, 9.628709865189816351865473415874, 10.87557325602807489897109934521, 11.50574266254959685851554612680, 12.46148392581718454359270192043
