| L(s) = 1 | − 2.37·2-s + 3-s + 3.64·4-s + 5-s − 2.37·6-s − 1.73·7-s − 3.91·8-s + 9-s − 2.37·10-s + 2.32·11-s + 3.64·12-s − 13-s + 4.11·14-s + 15-s + 2.01·16-s + 2.20·17-s − 2.37·18-s + 0.499·19-s + 3.64·20-s − 1.73·21-s − 5.52·22-s + 0.290·23-s − 3.91·24-s + 25-s + 2.37·26-s + 27-s − 6.31·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 0.577·3-s + 1.82·4-s + 0.447·5-s − 0.970·6-s − 0.654·7-s − 1.38·8-s + 0.333·9-s − 0.751·10-s + 0.701·11-s + 1.05·12-s − 0.277·13-s + 1.09·14-s + 0.258·15-s + 0.503·16-s + 0.533·17-s − 0.560·18-s + 0.114·19-s + 0.815·20-s − 0.377·21-s − 1.17·22-s + 0.0605·23-s − 0.799·24-s + 0.200·25-s + 0.466·26-s + 0.192·27-s − 1.19·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 - 2.32T + 11T^{2} \) |
| 17 | \( 1 - 2.20T + 17T^{2} \) |
| 19 | \( 1 - 0.499T + 19T^{2} \) |
| 23 | \( 1 - 0.290T + 23T^{2} \) |
| 29 | \( 1 + 9.49T + 29T^{2} \) |
| 37 | \( 1 - 4.13T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 + 2.64T + 43T^{2} \) |
| 47 | \( 1 + 5.28T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 - 2.99T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 + 4.82T + 71T^{2} \) |
| 73 | \( 1 + 6.04T + 73T^{2} \) |
| 79 | \( 1 - 1.83T + 79T^{2} \) |
| 83 | \( 1 + 3.98T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 8.58T + 97T^{2} \) |
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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
−7.75501739348705507006967755953, −7.34240679721577704728649055901, −6.54688554059671808842236963807, −5.99532733431075890188620664338, −4.87294604145270643554001283388, −3.69717461108229075284193159953, −2.95663049670082320578789285126, −1.98059395430779290012101433868, −1.30573694713550115863191049646, 0,
1.30573694713550115863191049646, 1.98059395430779290012101433868, 2.95663049670082320578789285126, 3.69717461108229075284193159953, 4.87294604145270643554001283388, 5.99532733431075890188620664338, 6.54688554059671808842236963807, 7.34240679721577704728649055901, 7.75501739348705507006967755953
