| L(s) = 1 | − 2.32·2-s + 3.39·4-s + 3.53·5-s − 1.40·7-s − 3.24·8-s − 8.21·10-s + 6.53·11-s − 3.29·13-s + 3.25·14-s + 0.745·16-s − 1.57·17-s + 5.79·19-s + 12.0·20-s − 15.1·22-s − 5.41·23-s + 7.51·25-s + 7.65·26-s − 4.76·28-s − 7.46·29-s + 4.41·31-s + 4.75·32-s + 3.67·34-s − 4.96·35-s + 5.73·37-s − 13.4·38-s − 11.4·40-s + 8.92·41-s + ⋯ |
| L(s) = 1 | − 1.64·2-s + 1.69·4-s + 1.58·5-s − 0.529·7-s − 1.14·8-s − 2.59·10-s + 1.97·11-s − 0.913·13-s + 0.870·14-s + 0.186·16-s − 0.383·17-s + 1.33·19-s + 2.68·20-s − 3.23·22-s − 1.12·23-s + 1.50·25-s + 1.50·26-s − 0.900·28-s − 1.38·29-s + 0.793·31-s + 0.841·32-s + 0.629·34-s − 0.838·35-s + 0.943·37-s − 2.18·38-s − 1.81·40-s + 1.39·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.264130655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.264130655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 5 | \( 1 - 3.53T + 5T^{2} \) |
| 7 | \( 1 + 1.40T + 7T^{2} \) |
| 11 | \( 1 - 6.53T + 11T^{2} \) |
| 13 | \( 1 + 3.29T + 13T^{2} \) |
| 17 | \( 1 + 1.57T + 17T^{2} \) |
| 19 | \( 1 - 5.79T + 19T^{2} \) |
| 23 | \( 1 + 5.41T + 23T^{2} \) |
| 29 | \( 1 + 7.46T + 29T^{2} \) |
| 31 | \( 1 - 4.41T + 31T^{2} \) |
| 37 | \( 1 - 5.73T + 37T^{2} \) |
| 41 | \( 1 - 8.92T + 41T^{2} \) |
| 43 | \( 1 + 1.70T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 2.49T + 53T^{2} \) |
| 59 | \( 1 + 2.99T + 59T^{2} \) |
| 61 | \( 1 - 8.38T + 61T^{2} \) |
| 67 | \( 1 - 7.33T + 67T^{2} \) |
| 71 | \( 1 - 1.30T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 4.15T + 79T^{2} \) |
| 83 | \( 1 - 4.33T + 83T^{2} \) |
| 89 | \( 1 - 2.44T + 89T^{2} \) |
| 97 | \( 1 - 1.32T + 97T^{2} \) |
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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
−8.193906741485835819189029153399, −7.46265936830650076014622230172, −6.60950706416707300546555521297, −6.37629253682787931366595771687, −5.55518598408230923752583861566, −4.46958677694924096183568125856, −3.32121655895964910841470239541, −2.24983890047930781805402364104, −1.69480053713141616228573972707, −0.78701295711334507448968969256,
0.78701295711334507448968969256, 1.69480053713141616228573972707, 2.24983890047930781805402364104, 3.32121655895964910841470239541, 4.46958677694924096183568125856, 5.55518598408230923752583861566, 6.37629253682787931366595771687, 6.60950706416707300546555521297, 7.46265936830650076014622230172, 8.193906741485835819189029153399
