| L(s) = 1 | − 4.68·2-s + 13.9·4-s + 9.12·5-s − 4.35·7-s − 27.6·8-s − 42.6·10-s − 0.264·11-s + 32.6·13-s + 20.4·14-s + 18.2·16-s − 37.6·17-s − 64.7·19-s + 126.·20-s + 1.23·22-s + 77.1·23-s − 41.8·25-s − 152.·26-s − 60.6·28-s + 194.·29-s − 176.·31-s + 135.·32-s + 176.·34-s − 39.7·35-s + 174.·37-s + 303.·38-s − 252.·40-s + 411.·41-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.73·4-s + 0.815·5-s − 0.235·7-s − 1.22·8-s − 1.35·10-s − 0.00725·11-s + 0.696·13-s + 0.389·14-s + 0.285·16-s − 0.536·17-s − 0.782·19-s + 1.41·20-s + 0.0120·22-s + 0.699·23-s − 0.334·25-s − 1.15·26-s − 0.409·28-s + 1.24·29-s − 1.02·31-s + 0.750·32-s + 0.888·34-s − 0.192·35-s + 0.775·37-s + 1.29·38-s − 0.998·40-s + 1.56·41-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 | \( 1 + 4.68T + 8T^{2} \) |
| 5 | \( 1 - 9.12T + 125T^{2} \) |
| 7 | \( 1 + 4.35T + 343T^{2} \) |
| 11 | \( 1 + 0.264T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 37.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 64.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 77.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 176.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 174.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 411.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 487.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 25.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 635.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 414.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 521.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 982.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 223.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 501.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 777.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 624.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 758.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 986.T + 9.12e5T^{2} \) |
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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
−8.470849682016241543129757655499, −7.87200162739593029065586853247, −6.72240190402112269278464135321, −6.46256448603352915052093324209, −5.41003272598611797206512774075, −4.20496200923687669334737409917, −2.86471687506458105455511011415, −1.98557505540619835062320150203, −1.13779903063930030055458367186, 0,
1.13779903063930030055458367186, 1.98557505540619835062320150203, 2.86471687506458105455511011415, 4.20496200923687669334737409917, 5.41003272598611797206512774075, 6.46256448603352915052093324209, 6.72240190402112269278464135321, 7.87200162739593029065586853247, 8.470849682016241543129757655499
