| L(s) = 1 | + 1.78·2-s − 3-s + 1.17·4-s − 1.78·6-s − 1.47·8-s + 9-s − 2.07·11-s − 1.17·12-s + 3.13·13-s − 4.96·16-s − 2.13·17-s + 1.78·18-s + 7.73·19-s − 3.70·22-s − 5.53·23-s + 1.47·24-s + 5.57·26-s − 27-s − 4.01·29-s − 2.91·31-s − 5.91·32-s + 2.07·33-s − 3.79·34-s + 1.17·36-s − 3.51·37-s + 13.7·38-s − 3.13·39-s + ⋯ |
| L(s) = 1 | + 1.25·2-s − 0.577·3-s + 0.587·4-s − 0.727·6-s − 0.520·8-s + 0.333·9-s − 0.626·11-s − 0.338·12-s + 0.868·13-s − 1.24·16-s − 0.516·17-s + 0.419·18-s + 1.77·19-s − 0.789·22-s − 1.15·23-s + 0.300·24-s + 1.09·26-s − 0.192·27-s − 0.745·29-s − 0.522·31-s − 1.04·32-s + 0.361·33-s − 0.650·34-s + 0.195·36-s − 0.577·37-s + 2.23·38-s − 0.501·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.78T + 2T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 - 3.13T + 13T^{2} \) |
| 17 | \( 1 + 2.13T + 17T^{2} \) |
| 19 | \( 1 - 7.73T + 19T^{2} \) |
| 23 | \( 1 + 5.53T + 23T^{2} \) |
| 29 | \( 1 + 4.01T + 29T^{2} \) |
| 31 | \( 1 + 2.91T + 31T^{2} \) |
| 37 | \( 1 + 3.51T + 37T^{2} \) |
| 41 | \( 1 + 7.99T + 41T^{2} \) |
| 43 | \( 1 - 4.99T + 43T^{2} \) |
| 47 | \( 1 - 2.44T + 47T^{2} \) |
| 53 | \( 1 + 9.91T + 53T^{2} \) |
| 59 | \( 1 + 2.95T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 3.83T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 + 8.55T + 73T^{2} \) |
| 79 | \( 1 - 8.11T + 79T^{2} \) |
| 83 | \( 1 + 8.75T + 83T^{2} \) |
| 89 | \( 1 + 0.618T + 89T^{2} \) |
| 97 | \( 1 + 0.296T + 97T^{2} \) |
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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
−8.000689943029824231238949838706, −7.19087719941793409954853967347, −6.40076271289141992994496614224, −5.60361803642221764505243467654, −5.31635403722634719792821361147, −4.34252913152126819795042220632, −3.65479918390440868576766641332, −2.84758462789998137917466138838, −1.60126592984958044405233691238, 0,
1.60126592984958044405233691238, 2.84758462789998137917466138838, 3.65479918390440868576766641332, 4.34252913152126819795042220632, 5.31635403722634719792821361147, 5.60361803642221764505243467654, 6.40076271289141992994496614224, 7.19087719941793409954853967347, 8.000689943029824231238949838706
