| L(s) = 1 | − 2.62·2-s − 0.813·3-s + 4.87·4-s + 0.562·5-s + 2.13·6-s − 5.01·7-s − 7.53·8-s − 2.33·9-s − 1.47·10-s − 2.09·11-s − 3.96·12-s − 13-s + 13.1·14-s − 0.457·15-s + 10.0·16-s + 3.74·17-s + 6.13·18-s − 0.888·19-s + 2.73·20-s + 4.07·21-s + 5.50·22-s + 3.39·23-s + 6.12·24-s − 4.68·25-s + 2.62·26-s + 4.34·27-s − 24.4·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 0.469·3-s + 2.43·4-s + 0.251·5-s + 0.870·6-s − 1.89·7-s − 2.66·8-s − 0.779·9-s − 0.466·10-s − 0.632·11-s − 1.14·12-s − 0.277·13-s + 3.51·14-s − 0.117·15-s + 2.50·16-s + 0.907·17-s + 1.44·18-s − 0.203·19-s + 0.612·20-s + 0.889·21-s + 1.17·22-s + 0.707·23-s + 1.25·24-s − 0.936·25-s + 0.514·26-s + 0.835·27-s − 4.61·28-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 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 + 0.813T + 3T^{2} \) |
| 5 | \( 1 - 0.562T + 5T^{2} \) |
| 7 | \( 1 + 5.01T + 7T^{2} \) |
| 11 | \( 1 + 2.09T + 11T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 + 0.888T + 19T^{2} \) |
| 23 | \( 1 - 3.39T + 23T^{2} \) |
| 29 | \( 1 - 3.62T + 29T^{2} \) |
| 31 | \( 1 + 8.47T + 31T^{2} \) |
| 37 | \( 1 - 2.48T + 37T^{2} \) |
| 41 | \( 1 + 7.93T + 41T^{2} \) |
| 43 | \( 1 - 6.00T + 43T^{2} \) |
| 47 | \( 1 - 1.50T + 47T^{2} \) |
| 53 | \( 1 + 2.70T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 0.919T + 61T^{2} \) |
| 67 | \( 1 + 2.20T + 67T^{2} \) |
| 71 | \( 1 - 1.75T + 71T^{2} \) |
| 73 | \( 1 - 9.37T + 73T^{2} \) |
| 79 | \( 1 - 6.77T + 79T^{2} \) |
| 83 | \( 1 - 8.69T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 6.56T + 97T^{2} \) |
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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
−7.78879271900984040018641101178, −7.11383717688115743273312525266, −6.50901118646526746648887273674, −5.89950319811614039285942140422, −5.31275233549278313272161522788, −3.58170062073974579636029611681, −2.90438685929736974340027631031, −2.19198063366144741338114308686, −0.77658123173333422057344773867, 0,
0.77658123173333422057344773867, 2.19198063366144741338114308686, 2.90438685929736974340027631031, 3.58170062073974579636029611681, 5.31275233549278313272161522788, 5.89950319811614039285942140422, 6.50901118646526746648887273674, 7.11383717688115743273312525266, 7.78879271900984040018641101178
