L(s) = 1 | − 2.59·2-s + 1.19·3-s + 4.73·4-s − 2.90·5-s − 3.10·6-s − 3.26·7-s − 7.11·8-s − 1.56·9-s + 7.53·10-s − 4.60·11-s + 5.67·12-s − 13-s + 8.48·14-s − 3.47·15-s + 8.98·16-s − 1.24·17-s + 4.06·18-s + 6.66·19-s − 13.7·20-s − 3.91·21-s + 11.9·22-s + 4.30·23-s − 8.51·24-s + 3.42·25-s + 2.59·26-s − 5.46·27-s − 15.4·28-s + ⋯ |
L(s) = 1 | − 1.83·2-s + 0.691·3-s + 2.36·4-s − 1.29·5-s − 1.26·6-s − 1.23·7-s − 2.51·8-s − 0.521·9-s + 2.38·10-s − 1.38·11-s + 1.63·12-s − 0.277·13-s + 2.26·14-s − 0.897·15-s + 2.24·16-s − 0.302·17-s + 0.957·18-s + 1.53·19-s − 3.07·20-s − 0.854·21-s + 2.54·22-s + 0.896·23-s − 1.73·24-s + 0.685·25-s + 0.509·26-s − 1.05·27-s − 2.92·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + T \) |
| 463 | \( 1 + T \) |
good | 2 | \( 1 + 2.59T + 2T^{2} \) |
| 3 | \( 1 - 1.19T + 3T^{2} \) |
| 5 | \( 1 + 2.90T + 5T^{2} \) |
| 7 | \( 1 + 3.26T + 7T^{2} \) |
| 11 | \( 1 + 4.60T + 11T^{2} \) |
| 17 | \( 1 + 1.24T + 17T^{2} \) |
| 19 | \( 1 - 6.66T + 19T^{2} \) |
| 23 | \( 1 - 4.30T + 23T^{2} \) |
| 29 | \( 1 - 1.29T + 29T^{2} \) |
| 31 | \( 1 - 7.84T + 31T^{2} \) |
| 37 | \( 1 - 0.108T + 37T^{2} \) |
| 41 | \( 1 + 0.176T + 41T^{2} \) |
| 43 | \( 1 - 9.16T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 + 1.97T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 4.22T + 71T^{2} \) |
| 73 | \( 1 + 8.85T + 73T^{2} \) |
| 79 | \( 1 - 1.58T + 79T^{2} \) |
| 83 | \( 1 - 9.08T + 83T^{2} \) |
| 89 | \( 1 + 8.90T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−7.74846675703317539324123838639, −7.52416147593117013436678383367, −6.76733858459908070484291192290, −5.91967452415949479549990582417, −4.85483945318090110530021282807, −3.45100802233732308279341082486, −2.98171059036725760456058704748, −2.42343063749998980957191836013, −0.814562074720338650154687767838, 0,
0.814562074720338650154687767838, 2.42343063749998980957191836013, 2.98171059036725760456058704748, 3.45100802233732308279341082486, 4.85483945318090110530021282807, 5.91967452415949479549990582417, 6.76733858459908070484291192290, 7.52416147593117013436678383367, 7.74846675703317539324123838639