| L(s) = 1 | + 5.33·2-s + 20.5·4-s + 19.5·5-s − 20.6·7-s + 66.7·8-s + 104.·10-s − 44.8·11-s − 0.493·13-s − 110.·14-s + 192.·16-s − 78.7·17-s − 26.1·19-s + 399.·20-s − 239.·22-s + 23·23-s + 255.·25-s − 2.63·26-s − 422.·28-s − 85.1·29-s − 4.79·31-s + 492.·32-s − 420.·34-s − 401.·35-s − 47.9·37-s − 139.·38-s + 1.30e3·40-s + 426.·41-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 2.56·4-s + 1.74·5-s − 1.11·7-s + 2.94·8-s + 3.29·10-s − 1.22·11-s − 0.0105·13-s − 2.10·14-s + 3.00·16-s − 1.12·17-s − 0.316·19-s + 4.47·20-s − 2.31·22-s + 0.208·23-s + 2.04·25-s − 0.0198·26-s − 2.85·28-s − 0.545·29-s − 0.0277·31-s + 2.72·32-s − 2.12·34-s − 1.94·35-s − 0.212·37-s − 0.596·38-s + 5.14·40-s + 1.62·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.954843536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.954843536\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 5.33T + 8T^{2} \) |
| 5 | \( 1 - 19.5T + 125T^{2} \) |
| 7 | \( 1 + 20.6T + 343T^{2} \) |
| 11 | \( 1 + 44.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.493T + 2.19e3T^{2} \) |
| 17 | \( 1 + 78.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.1T + 6.85e3T^{2} \) |
| 29 | \( 1 + 85.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 4.79T + 2.97e4T^{2} \) |
| 37 | \( 1 + 47.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 426.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 472.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 29.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 141.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 538.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 66.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 405.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 468.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 243.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 323.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.45e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 389.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 920.T + 9.12e5T^{2} \) |
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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
−12.73026383882221228978765037976, −11.03668255123451730830247049710, −10.34404871147648666261553810095, −9.241055828180501580844624728167, −7.25014026932080824957787275607, −6.19544262892206765400043931215, −5.72490971122620012953680959686, −4.56941454205786451787765856607, −2.94743227438865136820549254803, −2.16687754914804856008362858119,
2.16687754914804856008362858119, 2.94743227438865136820549254803, 4.56941454205786451787765856607, 5.72490971122620012953680959686, 6.19544262892206765400043931215, 7.25014026932080824957787275607, 9.241055828180501580844624728167, 10.34404871147648666261553810095, 11.03668255123451730830247049710, 12.73026383882221228978765037976
