| L(s) = 1 | − 1.72·3-s − 5-s − 7-s − 0.0236·9-s + 1.64·11-s + 6.74·13-s + 1.72·15-s + 7.32·17-s − 1.33·19-s + 1.72·21-s + 3.73·23-s + 25-s + 5.21·27-s + 3.06·29-s − 6.07·31-s − 2.83·33-s + 35-s + 3.80·37-s − 11.6·39-s − 0.284·41-s − 43-s + 0.0236·45-s − 2.36·47-s + 49-s − 12.6·51-s − 10.2·53-s − 1.64·55-s + ⋯ |
| L(s) = 1 | − 0.996·3-s − 0.447·5-s − 0.377·7-s − 0.00789·9-s + 0.495·11-s + 1.87·13-s + 0.445·15-s + 1.77·17-s − 0.306·19-s + 0.376·21-s + 0.778·23-s + 0.200·25-s + 1.00·27-s + 0.569·29-s − 1.09·31-s − 0.493·33-s + 0.169·35-s + 0.625·37-s − 1.86·39-s − 0.0443·41-s − 0.152·43-s + 0.00352·45-s − 0.344·47-s + 0.142·49-s − 1.77·51-s − 1.41·53-s − 0.221·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.330039602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.330039602\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 1.72T + 3T^{2} \) |
| 11 | \( 1 - 1.64T + 11T^{2} \) |
| 13 | \( 1 - 6.74T + 13T^{2} \) |
| 17 | \( 1 - 7.32T + 17T^{2} \) |
| 19 | \( 1 + 1.33T + 19T^{2} \) |
| 23 | \( 1 - 3.73T + 23T^{2} \) |
| 29 | \( 1 - 3.06T + 29T^{2} \) |
| 31 | \( 1 + 6.07T + 31T^{2} \) |
| 37 | \( 1 - 3.80T + 37T^{2} \) |
| 41 | \( 1 + 0.284T + 41T^{2} \) |
| 47 | \( 1 + 2.36T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 + 0.907T + 73T^{2} \) |
| 79 | \( 1 - 0.835T + 79T^{2} \) |
| 83 | \( 1 + 8.20T + 83T^{2} \) |
| 89 | \( 1 + 2.30T + 89T^{2} \) |
| 97 | \( 1 + 7.13T + 97T^{2} \) |
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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
−8.114710828608756247159246348859, −7.27077012837258497752648511379, −6.43417307261487743583036447812, −6.01812522460302076131565868856, −5.37327449748733546634338414860, −4.50214242539231325222518043002, −3.54588044101260793227668729196, −3.12538184577513037220684121064, −1.46541910082993085983775065307, −0.70289006196347506012214894561,
0.70289006196347506012214894561, 1.46541910082993085983775065307, 3.12538184577513037220684121064, 3.54588044101260793227668729196, 4.50214242539231325222518043002, 5.37327449748733546634338414860, 6.01812522460302076131565868856, 6.43417307261487743583036447812, 7.27077012837258497752648511379, 8.114710828608756247159246348859
