| L(s) = 1 | + 5.03·2-s + 17.3·4-s + 1.37·5-s − 13.6·7-s + 47.2·8-s + 6.92·10-s + 34.8·11-s − 80.5·13-s − 68.9·14-s + 99.1·16-s − 15.8·17-s − 143.·19-s + 23.8·20-s + 175.·22-s + 29.0·23-s − 123.·25-s − 405.·26-s − 237.·28-s + 6.80·29-s + 202.·31-s + 121.·32-s − 79.6·34-s − 18.8·35-s + 320.·37-s − 723.·38-s + 64.9·40-s − 271.·41-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 2.17·4-s + 0.122·5-s − 0.738·7-s + 2.08·8-s + 0.218·10-s + 0.955·11-s − 1.71·13-s − 1.31·14-s + 1.54·16-s − 0.225·17-s − 1.73·19-s + 0.267·20-s + 1.70·22-s + 0.263·23-s − 0.984·25-s − 3.06·26-s − 1.60·28-s + 0.0435·29-s + 1.17·31-s + 0.669·32-s − 0.401·34-s − 0.0908·35-s + 1.42·37-s − 3.09·38-s + 0.256·40-s − 1.03·41-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.03T + 8T^{2} \) |
| 5 | \( 1 - 1.37T + 125T^{2} \) |
| 7 | \( 1 + 13.6T + 343T^{2} \) |
| 11 | \( 1 - 34.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 80.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 15.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 143.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 29.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 6.80T + 2.43e4T^{2} \) |
| 31 | \( 1 - 202.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 320.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 271.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 64.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 359.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 311.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 219.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 434.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 33.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 898.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 315.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 423.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.35e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 720.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 57.4T + 9.12e5T^{2} \) |
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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
−8.085184580751760204750617474031, −7.12895375208278563062712110910, −6.38745197476099918967716847514, −6.10769326907653748485536752848, −4.80487615573940975229452735038, −4.46774643336311966219716872634, −3.47778340785871406992390477618, −2.64034501887332727503964031434, −1.83655992412120674801225063156, 0,
1.83655992412120674801225063156, 2.64034501887332727503964031434, 3.47778340785871406992390477618, 4.46774643336311966219716872634, 4.80487615573940975229452735038, 6.10769326907653748485536752848, 6.38745197476099918967716847514, 7.12895375208278563062712110910, 8.085184580751760204750617474031
