| L(s) = 1 | − 2-s + 3-s + 4-s − 3.05·5-s − 6-s − 2.36·7-s − 8-s + 9-s + 3.05·10-s + 2.94·11-s + 12-s + 0.467·13-s + 2.36·14-s − 3.05·15-s + 16-s − 1.03·17-s − 18-s − 19-s − 3.05·20-s − 2.36·21-s − 2.94·22-s + 5.42·23-s − 24-s + 4.33·25-s − 0.467·26-s + 27-s − 2.36·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.36·5-s − 0.408·6-s − 0.892·7-s − 0.353·8-s + 0.333·9-s + 0.966·10-s + 0.889·11-s + 0.288·12-s + 0.129·13-s + 0.630·14-s − 0.788·15-s + 0.250·16-s − 0.250·17-s − 0.235·18-s − 0.229·19-s − 0.683·20-s − 0.515·21-s − 0.628·22-s + 1.13·23-s − 0.204·24-s + 0.866·25-s − 0.0916·26-s + 0.192·27-s − 0.446·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 + 3.05T + 5T^{2} \) |
| 7 | \( 1 + 2.36T + 7T^{2} \) |
| 11 | \( 1 - 2.94T + 11T^{2} \) |
| 13 | \( 1 - 0.467T + 13T^{2} \) |
| 17 | \( 1 + 1.03T + 17T^{2} \) |
| 23 | \( 1 - 5.42T + 23T^{2} \) |
| 29 | \( 1 + 3.45T + 29T^{2} \) |
| 31 | \( 1 + 3.18T + 31T^{2} \) |
| 37 | \( 1 - 4.86T + 37T^{2} \) |
| 41 | \( 1 + 6.40T + 41T^{2} \) |
| 43 | \( 1 + 9.23T + 43T^{2} \) |
| 47 | \( 1 - 9.30T + 47T^{2} \) |
| 59 | \( 1 - 9.29T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 2.33T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 - 4.08T + 83T^{2} \) |
| 89 | \( 1 + 0.763T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−7.78033246988956652162339834385, −6.98632980007425684972937762581, −6.82093979128041433002456657133, −5.73903099047612029261242538546, −4.59286609766803646286342349580, −3.71035342811229255987508342415, −3.38558864553160487292757000919, −2.34766577553792807468768287774, −1.12094295127730239845931828915, 0,
1.12094295127730239845931828915, 2.34766577553792807468768287774, 3.38558864553160487292757000919, 3.71035342811229255987508342415, 4.59286609766803646286342349580, 5.73903099047612029261242538546, 6.82093979128041433002456657133, 6.98632980007425684972937762581, 7.78033246988956652162339834385
