| L(s) = 1 | − 2.66·2-s − 2.16·3-s + 5.10·4-s − 2.11·5-s + 5.75·6-s − 1.28·7-s − 8.26·8-s + 1.66·9-s + 5.64·10-s + 1.13·11-s − 11.0·12-s − 5.51·13-s + 3.43·14-s + 4.57·15-s + 11.8·16-s − 5.20·17-s − 4.44·18-s − 19-s − 10.8·20-s + 2.78·21-s − 3.02·22-s + 2.25·23-s + 17.8·24-s − 0.519·25-s + 14.7·26-s + 2.88·27-s − 6.57·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s − 1.24·3-s + 2.55·4-s − 0.946·5-s + 2.35·6-s − 0.486·7-s − 2.92·8-s + 0.555·9-s + 1.78·10-s + 0.341·11-s − 3.18·12-s − 1.52·13-s + 0.917·14-s + 1.18·15-s + 2.95·16-s − 1.26·17-s − 1.04·18-s − 0.229·19-s − 2.41·20-s + 0.607·21-s − 0.644·22-s + 0.469·23-s + 3.64·24-s − 0.103·25-s + 2.88·26-s + 0.554·27-s − 1.24·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + T \) |
| 317 | \( 1 + T \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 3 | \( 1 + 2.16T + 3T^{2} \) |
| 5 | \( 1 + 2.11T + 5T^{2} \) |
| 7 | \( 1 + 1.28T + 7T^{2} \) |
| 11 | \( 1 - 1.13T + 11T^{2} \) |
| 13 | \( 1 + 5.51T + 13T^{2} \) |
| 17 | \( 1 + 5.20T + 17T^{2} \) |
| 23 | \( 1 - 2.25T + 23T^{2} \) |
| 29 | \( 1 + 0.811T + 29T^{2} \) |
| 31 | \( 1 + 5.44T + 31T^{2} \) |
| 37 | \( 1 - 0.251T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 7.08T + 43T^{2} \) |
| 47 | \( 1 + 3.71T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 9.55T + 59T^{2} \) |
| 61 | \( 1 + 3.20T + 61T^{2} \) |
| 67 | \( 1 + 2.36T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 9.77T + 83T^{2} \) |
| 89 | \( 1 - 3.64T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−7.71000078897384258690086242453, −7.02382557175135028483843192011, −6.73814214861148823918923527025, −5.91396486152506342817503108973, −5.01388546916528173882204557676, −4.03919336652050297761654137534, −2.85185476834098826668489401943, −1.98074091194267514906416822901, −0.63034672079409015445642355079, 0,
0.63034672079409015445642355079, 1.98074091194267514906416822901, 2.85185476834098826668489401943, 4.03919336652050297761654137534, 5.01388546916528173882204557676, 5.91396486152506342817503108973, 6.73814214861148823918923527025, 7.02382557175135028483843192011, 7.71000078897384258690086242453
