| L(s) = 1 | − 1.64·2-s + 3-s + 0.708·4-s − 0.419·5-s − 1.64·6-s + 1.49·7-s + 2.12·8-s + 9-s + 0.690·10-s − 2.79·11-s + 0.708·12-s − 0.608·13-s − 2.45·14-s − 0.419·15-s − 4.91·16-s + 17-s − 1.64·18-s + 4.76·19-s − 0.297·20-s + 1.49·21-s + 4.60·22-s − 1.47·23-s + 2.12·24-s − 4.82·25-s + 1.00·26-s + 27-s + 1.05·28-s + ⋯ |
| L(s) = 1 | − 1.16·2-s + 0.577·3-s + 0.354·4-s − 0.187·5-s − 0.671·6-s + 0.563·7-s + 0.751·8-s + 0.333·9-s + 0.218·10-s − 0.843·11-s + 0.204·12-s − 0.168·13-s − 0.656·14-s − 0.108·15-s − 1.22·16-s + 0.242·17-s − 0.387·18-s + 1.09·19-s − 0.0664·20-s + 0.325·21-s + 0.981·22-s − 0.306·23-s + 0.433·24-s − 0.964·25-s + 0.196·26-s + 0.192·27-s + 0.199·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 1.64T + 2T^{2} \) |
| 5 | \( 1 + 0.419T + 5T^{2} \) |
| 7 | \( 1 - 1.49T + 7T^{2} \) |
| 11 | \( 1 + 2.79T + 11T^{2} \) |
| 13 | \( 1 + 0.608T + 13T^{2} \) |
| 19 | \( 1 - 4.76T + 19T^{2} \) |
| 23 | \( 1 + 1.47T + 23T^{2} \) |
| 29 | \( 1 + 5.97T + 29T^{2} \) |
| 31 | \( 1 + 0.192T + 31T^{2} \) |
| 37 | \( 1 - 0.227T + 37T^{2} \) |
| 41 | \( 1 + 3.88T + 41T^{2} \) |
| 43 | \( 1 - 7.71T + 43T^{2} \) |
| 47 | \( 1 + 1.89T + 47T^{2} \) |
| 53 | \( 1 + 8.79T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 6.94T + 61T^{2} \) |
| 67 | \( 1 - 3.16T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 83 | \( 1 + 4.68T + 83T^{2} \) |
| 89 | \( 1 - 3.87T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−8.067113919854068279559990527826, −7.61790739636539164693741552375, −7.17811840439629568949839177439, −5.86219228674825737901255206765, −5.04347461211319906647987236620, −4.23901628744742153619178662364, −3.25674459737067913375115192629, −2.18223215786487754769637094327, −1.34919403430304579290362026022, 0,
1.34919403430304579290362026022, 2.18223215786487754769637094327, 3.25674459737067913375115192629, 4.23901628744742153619178662364, 5.04347461211319906647987236620, 5.86219228674825737901255206765, 7.17811840439629568949839177439, 7.61790739636539164693741552375, 8.067113919854068279559990527826
