| L(s) = 1 | − 2.03·2-s − 3-s + 2.12·4-s + 1.66·5-s + 2.03·6-s − 1.52·7-s − 0.260·8-s + 9-s − 3.39·10-s − 3.97·11-s − 2.12·12-s − 5.52·13-s + 3.09·14-s − 1.66·15-s − 3.72·16-s − 17-s − 2.03·18-s − 3.49·19-s + 3.55·20-s + 1.52·21-s + 8.07·22-s − 6.50·23-s + 0.260·24-s − 2.21·25-s + 11.2·26-s − 27-s − 3.24·28-s + ⋯ |
| L(s) = 1 | − 1.43·2-s − 0.577·3-s + 1.06·4-s + 0.746·5-s + 0.829·6-s − 0.576·7-s − 0.0921·8-s + 0.333·9-s − 1.07·10-s − 1.19·11-s − 0.614·12-s − 1.53·13-s + 0.827·14-s − 0.431·15-s − 0.931·16-s − 0.242·17-s − 0.478·18-s − 0.802·19-s + 0.794·20-s + 0.332·21-s + 1.72·22-s − 1.35·23-s + 0.0531·24-s − 0.442·25-s + 2.20·26-s − 0.192·27-s − 0.613·28-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
| \(L(1)\) |
\(\approx\) |
\(0.1328842834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1328842834\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.03T + 2T^{2} \) |
| 5 | \( 1 - 1.66T + 5T^{2} \) |
| 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 + 3.97T + 11T^{2} \) |
| 13 | \( 1 + 5.52T + 13T^{2} \) |
| 19 | \( 1 + 3.49T + 19T^{2} \) |
| 23 | \( 1 + 6.50T + 23T^{2} \) |
| 29 | \( 1 + 9.08T + 29T^{2} \) |
| 31 | \( 1 + 3.43T + 31T^{2} \) |
| 37 | \( 1 + 0.355T + 37T^{2} \) |
| 41 | \( 1 + 8.42T + 41T^{2} \) |
| 43 | \( 1 + 6.82T + 43T^{2} \) |
| 47 | \( 1 - 13.4T + 47T^{2} \) |
| 53 | \( 1 + 9.28T + 53T^{2} \) |
| 59 | \( 1 - 3.98T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 + 6.77T + 89T^{2} \) |
| 97 | \( 1 + 4.06T + 97T^{2} \) |
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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
−8.433285614591560330833594328169, −7.79472509156870428946878670958, −7.12647964950674870442615207679, −6.46344357534835942241938735902, −5.54667401812346088167047852181, −4.96692409906376985753697001214, −3.79567620708206109592874083512, −2.25599040864860696933186118266, −2.01591522951575852108814220595, −0.24769022010946652530868098900,
0.24769022010946652530868098900, 2.01591522951575852108814220595, 2.25599040864860696933186118266, 3.79567620708206109592874083512, 4.96692409906376985753697001214, 5.54667401812346088167047852181, 6.46344357534835942241938735902, 7.12647964950674870442615207679, 7.79472509156870428946878670958, 8.433285614591560330833594328169
