| L(s) = 1 | − 0.363·2-s − 1.86·4-s + 0.137·5-s + 3.90·7-s + 1.40·8-s − 0.0501·10-s + 11-s − 3.52·13-s − 1.42·14-s + 3.22·16-s − 7.69·17-s − 0.962·19-s − 0.257·20-s − 0.363·22-s + 2.43·23-s − 4.98·25-s + 1.28·26-s − 7.29·28-s + 8.63·29-s − 9.23·31-s − 3.98·32-s + 2.79·34-s + 0.538·35-s + 7.50·37-s + 0.350·38-s + 0.194·40-s + 3.88·41-s + ⋯ |
| L(s) = 1 | − 0.257·2-s − 0.933·4-s + 0.0616·5-s + 1.47·7-s + 0.497·8-s − 0.0158·10-s + 0.301·11-s − 0.977·13-s − 0.379·14-s + 0.805·16-s − 1.86·17-s − 0.220·19-s − 0.0575·20-s − 0.0775·22-s + 0.508·23-s − 0.996·25-s + 0.251·26-s − 1.37·28-s + 1.60·29-s − 1.65·31-s − 0.705·32-s + 0.480·34-s + 0.0910·35-s + 1.23·37-s + 0.0568·38-s + 0.0306·40-s + 0.607·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.369441112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.369441112\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 0.363T + 2T^{2} \) |
| 5 | \( 1 - 0.137T + 5T^{2} \) |
| 7 | \( 1 - 3.90T + 7T^{2} \) |
| 13 | \( 1 + 3.52T + 13T^{2} \) |
| 17 | \( 1 + 7.69T + 17T^{2} \) |
| 19 | \( 1 + 0.962T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 - 8.63T + 29T^{2} \) |
| 31 | \( 1 + 9.23T + 31T^{2} \) |
| 37 | \( 1 - 7.50T + 37T^{2} \) |
| 41 | \( 1 - 3.88T + 41T^{2} \) |
| 43 | \( 1 - 5.36T + 43T^{2} \) |
| 47 | \( 1 + 4.27T + 47T^{2} \) |
| 53 | \( 1 - 6.82T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 3.34T + 71T^{2} \) |
| 73 | \( 1 - 2.44T + 73T^{2} \) |
| 79 | \( 1 + 9.20T + 79T^{2} \) |
| 83 | \( 1 + 3.69T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−8.340991967486372296749326239132, −7.45219261951948069370672492576, −6.87376903530995138380219166768, −5.75212450003324041035532772619, −5.06420762264564863518891742042, −4.41396113826473808393045126214, −4.04166780194901277633847250252, −2.54594063012275198084762645307, −1.80660365105223148398559272053, −0.65218050875753454375545420792,
0.65218050875753454375545420792, 1.80660365105223148398559272053, 2.54594063012275198084762645307, 4.04166780194901277633847250252, 4.41396113826473808393045126214, 5.06420762264564863518891742042, 5.75212450003324041035532772619, 6.87376903530995138380219166768, 7.45219261951948069370672492576, 8.340991967486372296749326239132
