| L(s) = 1 | + 1.36·2-s − 3-s − 0.149·4-s − 3.22·5-s − 1.36·6-s + 2.99·7-s − 2.92·8-s + 9-s − 4.39·10-s − 3.10·11-s + 0.149·12-s + 1.97·13-s + 4.07·14-s + 3.22·15-s − 3.67·16-s − 17-s + 1.36·18-s − 0.710·19-s + 0.483·20-s − 2.99·21-s − 4.21·22-s + 6.91·23-s + 2.92·24-s + 5.42·25-s + 2.69·26-s − 27-s − 0.449·28-s + ⋯ |
| L(s) = 1 | + 0.961·2-s − 0.577·3-s − 0.0749·4-s − 1.44·5-s − 0.555·6-s + 1.13·7-s − 1.03·8-s + 0.333·9-s − 1.38·10-s − 0.934·11-s + 0.0432·12-s + 0.549·13-s + 1.08·14-s + 0.833·15-s − 0.919·16-s − 0.242·17-s + 0.320·18-s − 0.163·19-s + 0.108·20-s − 0.653·21-s − 0.899·22-s + 1.44·23-s + 0.596·24-s + 1.08·25-s + 0.528·26-s − 0.192·27-s − 0.0848·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 1.36T + 2T^{2} \) |
| 5 | \( 1 + 3.22T + 5T^{2} \) |
| 7 | \( 1 - 2.99T + 7T^{2} \) |
| 11 | \( 1 + 3.10T + 11T^{2} \) |
| 13 | \( 1 - 1.97T + 13T^{2} \) |
| 19 | \( 1 + 0.710T + 19T^{2} \) |
| 23 | \( 1 - 6.91T + 23T^{2} \) |
| 29 | \( 1 - 9.35T + 29T^{2} \) |
| 31 | \( 1 + 7.97T + 31T^{2} \) |
| 37 | \( 1 + 4.06T + 37T^{2} \) |
| 41 | \( 1 + 0.959T + 41T^{2} \) |
| 43 | \( 1 + 1.47T + 43T^{2} \) |
| 47 | \( 1 - 5.23T + 47T^{2} \) |
| 53 | \( 1 - 4.09T + 53T^{2} \) |
| 59 | \( 1 - 2.47T + 59T^{2} \) |
| 61 | \( 1 - 8.55T + 61T^{2} \) |
| 67 | \( 1 + 5.39T + 67T^{2} \) |
| 71 | \( 1 + 0.392T + 71T^{2} \) |
| 73 | \( 1 + 3.98T + 73T^{2} \) |
| 79 | \( 1 + 5.00T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 9.10T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−7.33932062400207623165141215093, −6.84158216366765407659934038811, −5.82670491604275361720728763269, −5.06907849221385189041977630410, −4.80429562004053498327370913379, −4.05533224488613150701577736425, −3.42826714891881909018240378314, −2.51030955344667695255439805346, −1.08059591528998224626007079922, 0,
1.08059591528998224626007079922, 2.51030955344667695255439805346, 3.42826714891881909018240378314, 4.05533224488613150701577736425, 4.80429562004053498327370913379, 5.06907849221385189041977630410, 5.82670491604275361720728763269, 6.84158216366765407659934038811, 7.33932062400207623165141215093
