| L(s) = 1 | + 4.74·3-s − 5·5-s − 29.3·7-s − 4.44·9-s + 38.1·11-s − 22.5·13-s − 23.7·15-s − 104.·17-s − 141.·19-s − 139.·21-s + 23·23-s + 25·25-s − 149.·27-s + 241.·29-s − 99.2·31-s + 181.·33-s + 146.·35-s + 59.9·37-s − 107.·39-s + 249.·41-s + 163.·43-s + 22.2·45-s + 205.·47-s + 519.·49-s − 494.·51-s + 491.·53-s − 190.·55-s + ⋯ |
| L(s) = 1 | + 0.913·3-s − 0.447·5-s − 1.58·7-s − 0.164·9-s + 1.04·11-s − 0.480·13-s − 0.408·15-s − 1.48·17-s − 1.71·19-s − 1.44·21-s + 0.208·23-s + 0.200·25-s − 1.06·27-s + 1.54·29-s − 0.575·31-s + 0.956·33-s + 0.709·35-s + 0.266·37-s − 0.439·39-s + 0.949·41-s + 0.581·43-s + 0.0737·45-s + 0.638·47-s + 1.51·49-s − 1.35·51-s + 1.27·53-s − 0.467·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.362787026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.362787026\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 4.74T + 27T^{2} \) |
| 7 | \( 1 + 29.3T + 343T^{2} \) |
| 11 | \( 1 - 38.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 141.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 241.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 99.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 59.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 249.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 163.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 205.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 491.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 433.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 660.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 323.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 893.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 196.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 500.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 800.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 729.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.13e3T + 9.12e5T^{2} \) |
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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
−8.906929279719234184054463220403, −8.387952682218810126843147334615, −7.19517241691718878274655440259, −6.61730777380239690932254572626, −5.96009959231081196906180317465, −4.38809472123368319219420408334, −3.88431606693556110938599595462, −2.87092113838626018366321682818, −2.23482705976915827804534882134, −0.48929196218766658848787034185,
0.48929196218766658848787034185, 2.23482705976915827804534882134, 2.87092113838626018366321682818, 3.88431606693556110938599595462, 4.38809472123368319219420408334, 5.96009959231081196906180317465, 6.61730777380239690932254572626, 7.19517241691718878274655440259, 8.387952682218810126843147334615, 8.906929279719234184054463220403
