| L(s) = 1 | − 5.21·2-s + 11.2·3-s − 4.81·4-s − 58.7·6-s + 11.1·7-s + 191.·8-s − 116.·9-s − 557.·11-s − 54.2·12-s − 107.·13-s − 57.8·14-s − 846.·16-s − 329.·17-s + 604.·18-s + 2.93e3·19-s + 125.·21-s + 2.90e3·22-s + 385.·23-s + 2.16e3·24-s + 561.·26-s − 4.04e3·27-s − 53.4·28-s − 3.30e3·29-s − 5.47e3·31-s − 1.72e3·32-s − 6.27e3·33-s + 1.71e3·34-s + ⋯ |
| L(s) = 1 | − 0.921·2-s + 0.722·3-s − 0.150·4-s − 0.666·6-s + 0.0856·7-s + 1.06·8-s − 0.477·9-s − 1.38·11-s − 0.108·12-s − 0.176·13-s − 0.0789·14-s − 0.826·16-s − 0.276·17-s + 0.440·18-s + 1.86·19-s + 0.0619·21-s + 1.27·22-s + 0.151·23-s + 0.766·24-s + 0.162·26-s − 1.06·27-s − 0.0128·28-s − 0.730·29-s − 1.02·31-s − 0.298·32-s − 1.00·33-s + 0.254·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8193712727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8193712727\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 2 | \( 1 + 5.21T + 32T^{2} \) |
| 3 | \( 1 - 11.2T + 243T^{2} \) |
| 7 | \( 1 - 11.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 557.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 107.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 329.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.93e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 385.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.30e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.83e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.06e3T + 1.15e8T^{2} \) |
| 47 | \( 1 - 8.99e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.02e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.74e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.62e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.51e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 9.31e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.85e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.28e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 231.T + 8.58e9T^{2} \) |
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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
−9.297225123935295256946985647787, −8.293781631491168414071032235023, −7.77976853910727700713869582167, −7.17253410652911981770572858662, −5.57909765391498130342910275661, −4.99016781855547425703389642776, −3.65510275846412593336167374305, −2.73316253896737382886129574246, −1.70564341534441970953751094288, −0.42769636248817811913822015008,
0.42769636248817811913822015008, 1.70564341534441970953751094288, 2.73316253896737382886129574246, 3.65510275846412593336167374305, 4.99016781855547425703389642776, 5.57909765391498130342910275661, 7.17253410652911981770572858662, 7.77976853910727700713869582167, 8.293781631491168414071032235023, 9.297225123935295256946985647787
