| L(s) = 1 | + 3.82·2-s + 6.63·4-s − 13.2·5-s − 25.5·7-s − 5.22·8-s − 50.6·10-s − 38.8·11-s + 92.3·13-s − 97.8·14-s − 73.0·16-s − 10.1·17-s − 141.·19-s − 87.8·20-s − 148.·22-s − 180.·23-s + 50.4·25-s + 353.·26-s − 169.·28-s − 31.2·29-s + 92.0·31-s − 237.·32-s − 38.7·34-s + 338.·35-s + 308.·37-s − 539.·38-s + 69.1·40-s − 131.·41-s + ⋯ |
| L(s) = 1 | + 1.35·2-s + 0.829·4-s − 1.18·5-s − 1.38·7-s − 0.230·8-s − 1.60·10-s − 1.06·11-s + 1.97·13-s − 1.86·14-s − 1.14·16-s − 0.144·17-s − 1.70·19-s − 0.982·20-s − 1.44·22-s − 1.63·23-s + 0.403·25-s + 2.66·26-s − 1.14·28-s − 0.200·29-s + 0.533·31-s − 1.31·32-s − 0.195·34-s + 1.63·35-s + 1.36·37-s − 2.30·38-s + 0.273·40-s − 0.502·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.186684580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.186684580\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 3.82T + 8T^{2} \) |
| 5 | \( 1 + 13.2T + 125T^{2} \) |
| 7 | \( 1 + 25.5T + 343T^{2} \) |
| 11 | \( 1 + 38.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 92.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 141.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 31.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 92.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 308.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 131.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 97.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 148.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 67.0T + 1.48e5T^{2} \) |
| 59 | \( 1 - 220.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 662.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 134.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 229.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 16.2T + 3.89e5T^{2} \) |
| 79 | \( 1 - 996.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 591.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.45e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 236.T + 9.12e5T^{2} \) |
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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
−8.420373516390937363088769493685, −8.071149269833342630255512147318, −6.75469918769911445749125027938, −6.24614926128426273809921269476, −5.63874330807629691713249135032, −4.32480500318895056192622458030, −3.92400238624868808322547951371, −3.27668712220923526506741014562, −2.31850314987466863585231281692, −0.37426501933372017085798378537,
0.37426501933372017085798378537, 2.31850314987466863585231281692, 3.27668712220923526506741014562, 3.92400238624868808322547951371, 4.32480500318895056192622458030, 5.63874330807629691713249135032, 6.24614926128426273809921269476, 6.75469918769911445749125027938, 8.071149269833342630255512147318, 8.420373516390937363088769493685
