| L(s) = 1 | − 2.73·3-s − 5-s − 7-s + 4.46·9-s − 11-s − 1.46·13-s + 2.73·15-s − 3.46·17-s − 6.73·19-s + 2.73·21-s + 8.19·23-s + 25-s − 3.99·27-s − 4.73·29-s − 2·31-s + 2.73·33-s + 35-s + 0.732·37-s + 4·39-s − 2.19·41-s − 2·43-s − 4.46·45-s − 6.92·47-s + 49-s + 9.46·51-s − 7.26·53-s + 55-s + ⋯ |
| L(s) = 1 | − 1.57·3-s − 0.447·5-s − 0.377·7-s + 1.48·9-s − 0.301·11-s − 0.406·13-s + 0.705·15-s − 0.840·17-s − 1.54·19-s + 0.596·21-s + 1.70·23-s + 0.200·25-s − 0.769·27-s − 0.878·29-s − 0.359·31-s + 0.475·33-s + 0.169·35-s + 0.120·37-s + 0.640·39-s − 0.342·41-s − 0.304·43-s − 0.665·45-s − 1.01·47-s + 0.142·49-s + 1.32·51-s − 0.998·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2664885750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2664885750\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 6.73T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 0.732T + 37T^{2} \) |
| 41 | \( 1 + 2.19T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 7.26T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−7.88564588145823208291388660786, −7.11166102552308387219708627411, −6.54884408557104318721660470282, −6.05801450004200455731709333561, −5.03997177873655003403914594033, −4.74744442373279420810105582594, −3.83952629954115571878934888528, −2.78787195514632618009735907954, −1.61313355181238393416060098770, −0.29080531880755536608360242344,
0.29080531880755536608360242344, 1.61313355181238393416060098770, 2.78787195514632618009735907954, 3.83952629954115571878934888528, 4.74744442373279420810105582594, 5.03997177873655003403914594033, 6.05801450004200455731709333561, 6.54884408557104318721660470282, 7.11166102552308387219708627411, 7.88564588145823208291388660786
