| L(s) = 1 | + 2.57·2-s − 3-s + 4.60·4-s + 4.00·5-s − 2.57·6-s + 2.04·7-s + 6.70·8-s + 9-s + 10.3·10-s − 11-s − 4.60·12-s − 4.75·13-s + 5.25·14-s − 4.00·15-s + 8.02·16-s − 0.339·17-s + 2.57·18-s + 4.54·19-s + 18.4·20-s − 2.04·21-s − 2.57·22-s − 4.75·23-s − 6.70·24-s + 11.0·25-s − 12.2·26-s − 27-s + 9.41·28-s + ⋯ |
| L(s) = 1 | + 1.81·2-s − 0.577·3-s + 2.30·4-s + 1.79·5-s − 1.04·6-s + 0.772·7-s + 2.37·8-s + 0.333·9-s + 3.25·10-s − 0.301·11-s − 1.33·12-s − 1.31·13-s + 1.40·14-s − 1.03·15-s + 2.00·16-s − 0.0823·17-s + 0.605·18-s + 1.04·19-s + 4.13·20-s − 0.445·21-s − 0.548·22-s − 0.992·23-s − 1.36·24-s + 2.21·25-s − 2.39·26-s − 0.192·27-s + 1.77·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.290975298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.290975298\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.57T + 2T^{2} \) |
| 5 | \( 1 - 4.00T + 5T^{2} \) |
| 7 | \( 1 - 2.04T + 7T^{2} \) |
| 13 | \( 1 + 4.75T + 13T^{2} \) |
| 17 | \( 1 + 0.339T + 17T^{2} \) |
| 19 | \( 1 - 4.54T + 19T^{2} \) |
| 23 | \( 1 + 4.75T + 23T^{2} \) |
| 29 | \( 1 + 4.28T + 29T^{2} \) |
| 31 | \( 1 - 1.81T + 31T^{2} \) |
| 37 | \( 1 + 2.11T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 9.89T + 43T^{2} \) |
| 47 | \( 1 + 5.22T + 47T^{2} \) |
| 53 | \( 1 - 9.68T + 53T^{2} \) |
| 59 | \( 1 + 2.04T + 59T^{2} \) |
| 67 | \( 1 - 5.30T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 4.51T + 73T^{2} \) |
| 79 | \( 1 - 5.49T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 6.05T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.539870157825300915270752521735, −8.088814051518508490311618786710, −7.05609195986023363279029729472, −6.50091028228896444293205931826, −5.55688808400705891335623320882, −5.21690114532607586506612170266, −4.70513063402114353063060443086, −3.36752952133178251865428002151, −2.26028818187462232080986188574, −1.71024710475491943085368634998,
1.71024710475491943085368634998, 2.26028818187462232080986188574, 3.36752952133178251865428002151, 4.70513063402114353063060443086, 5.21690114532607586506612170266, 5.55688808400705891335623320882, 6.50091028228896444293205931826, 7.05609195986023363279029729472, 8.088814051518508490311618786710, 9.539870157825300915270752521735
