| L(s) = 1 | − 0.0473·2-s + 3.03·3-s − 1.99·4-s + 5-s − 0.143·6-s − 2.32·7-s + 0.189·8-s + 6.22·9-s − 0.0473·10-s − 4.55·11-s − 6.06·12-s − 3.49·13-s + 0.110·14-s + 3.03·15-s + 3.98·16-s − 17-s − 0.294·18-s + 6.86·19-s − 1.99·20-s − 7.06·21-s + 0.215·22-s − 3.26·23-s + 0.574·24-s + 25-s + 0.165·26-s + 9.80·27-s + 4.64·28-s + ⋯ |
| L(s) = 1 | − 0.0334·2-s + 1.75·3-s − 0.998·4-s + 0.447·5-s − 0.0586·6-s − 0.879·7-s + 0.0668·8-s + 2.07·9-s − 0.0149·10-s − 1.37·11-s − 1.75·12-s − 0.969·13-s + 0.0294·14-s + 0.784·15-s + 0.996·16-s − 0.242·17-s − 0.0694·18-s + 1.57·19-s − 0.446·20-s − 1.54·21-s + 0.0459·22-s − 0.680·23-s + 0.117·24-s + 0.200·25-s + 0.0324·26-s + 1.88·27-s + 0.878·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 + 0.0473T + 2T^{2} \) |
| 3 | \( 1 - 3.03T + 3T^{2} \) |
| 7 | \( 1 + 2.32T + 7T^{2} \) |
| 11 | \( 1 + 4.55T + 11T^{2} \) |
| 13 | \( 1 + 3.49T + 13T^{2} \) |
| 19 | \( 1 - 6.86T + 19T^{2} \) |
| 23 | \( 1 + 3.26T + 23T^{2} \) |
| 29 | \( 1 - 9.04T + 29T^{2} \) |
| 31 | \( 1 - 0.758T + 31T^{2} \) |
| 37 | \( 1 + 7.60T + 37T^{2} \) |
| 41 | \( 1 + 8.61T + 41T^{2} \) |
| 43 | \( 1 + 0.884T + 43T^{2} \) |
| 47 | \( 1 + 4.75T + 47T^{2} \) |
| 53 | \( 1 + 3.93T + 53T^{2} \) |
| 59 | \( 1 + 3.82T + 59T^{2} \) |
| 61 | \( 1 + 3.77T + 61T^{2} \) |
| 67 | \( 1 - 0.353T + 67T^{2} \) |
| 73 | \( 1 + 0.916T + 73T^{2} \) |
| 79 | \( 1 + 2.50T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 5.20T + 89T^{2} \) |
| 97 | \( 1 - 0.787T + 97T^{2} \) |
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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
−8.000663667321412393143322716925, −7.27909768887392339225433749093, −6.48077252875538282115633352210, −5.22888694461558962447002916268, −4.88028893195795137602204711305, −3.80318761921078709886213404022, −3.03694769432106645413644704458, −2.69260726989686914471780812952, −1.52429138168988951064395144683, 0,
1.52429138168988951064395144683, 2.69260726989686914471780812952, 3.03694769432106645413644704458, 3.80318761921078709886213404022, 4.88028893195795137602204711305, 5.22888694461558962447002916268, 6.48077252875538282115633352210, 7.27909768887392339225433749093, 8.000663667321412393143322716925
