| L(s) = 1 | − 0.987·2-s + 3-s − 1.02·4-s + 4.03·5-s − 0.987·6-s − 3.85·7-s + 2.98·8-s + 9-s − 3.98·10-s − 1.78·11-s − 1.02·12-s − 4.61·13-s + 3.80·14-s + 4.03·15-s − 0.901·16-s − 4.11·17-s − 0.987·18-s + 4.62·19-s − 4.13·20-s − 3.85·21-s + 1.75·22-s + 0.470·23-s + 2.98·24-s + 11.2·25-s + 4.55·26-s + 27-s + 3.94·28-s + ⋯ |
| L(s) = 1 | − 0.698·2-s + 0.577·3-s − 0.512·4-s + 1.80·5-s − 0.403·6-s − 1.45·7-s + 1.05·8-s + 0.333·9-s − 1.26·10-s − 0.537·11-s − 0.295·12-s − 1.27·13-s + 1.01·14-s + 1.04·15-s − 0.225·16-s − 0.997·17-s − 0.232·18-s + 1.06·19-s − 0.924·20-s − 0.841·21-s + 0.375·22-s + 0.0980·23-s + 0.609·24-s + 2.25·25-s + 0.893·26-s + 0.192·27-s + 0.746·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 0.987T + 2T^{2} \) |
| 5 | \( 1 - 4.03T + 5T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 11 | \( 1 + 1.78T + 11T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 + 4.11T + 17T^{2} \) |
| 19 | \( 1 - 4.62T + 19T^{2} \) |
| 23 | \( 1 - 0.470T + 23T^{2} \) |
| 29 | \( 1 - 1.10T + 29T^{2} \) |
| 37 | \( 1 - 0.323T + 37T^{2} \) |
| 41 | \( 1 + 7.25T + 41T^{2} \) |
| 43 | \( 1 - 0.000843T + 43T^{2} \) |
| 47 | \( 1 + 3.77T + 47T^{2} \) |
| 53 | \( 1 + 8.54T + 53T^{2} \) |
| 59 | \( 1 - 1.23T + 59T^{2} \) |
| 61 | \( 1 + 15.0T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 2.61T + 71T^{2} \) |
| 73 | \( 1 + 9.08T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 9.78T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 0.495T + 97T^{2} \) |
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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
−8.744266384466744051149811447983, −7.70283790907792230534990885020, −6.93789225212658866044835130745, −6.24716175906069428906195945642, −5.27507552746104539549916322012, −4.64532529107046618287683574726, −3.21472687784223564548107001990, −2.53460477986855324169959276894, −1.54621718659250880774506339691, 0,
1.54621718659250880774506339691, 2.53460477986855324169959276894, 3.21472687784223564548107001990, 4.64532529107046618287683574726, 5.27507552746104539549916322012, 6.24716175906069428906195945642, 6.93789225212658866044835130745, 7.70283790907792230534990885020, 8.744266384466744051149811447983
