| L(s) = 1 | + 8.07·2-s + 33.2·4-s + 25·5-s − 39.3·7-s + 10.2·8-s + 201.·10-s − 121·11-s − 220.·13-s − 318.·14-s − 981.·16-s + 200.·17-s + 350.·19-s + 831.·20-s − 977.·22-s − 1.38e3·23-s + 625·25-s − 1.78e3·26-s − 1.30e3·28-s − 5.50e3·29-s − 2.45e3·31-s − 8.25e3·32-s + 1.61e3·34-s − 984.·35-s − 4.06e3·37-s + 2.83e3·38-s + 255.·40-s − 527.·41-s + ⋯ |
| L(s) = 1 | + 1.42·2-s + 1.03·4-s + 0.447·5-s − 0.303·7-s + 0.0564·8-s + 0.638·10-s − 0.301·11-s − 0.362·13-s − 0.433·14-s − 0.958·16-s + 0.168·17-s + 0.222·19-s + 0.464·20-s − 0.430·22-s − 0.545·23-s + 0.200·25-s − 0.517·26-s − 0.315·28-s − 1.21·29-s − 0.458·31-s − 1.42·32-s + 0.240·34-s − 0.135·35-s − 0.487·37-s + 0.317·38-s + 0.0252·40-s − 0.0489·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 8.07T + 32T^{2} \) |
| 7 | \( 1 + 39.3T + 1.68e4T^{2} \) |
| 13 | \( 1 + 220.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 200.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 350.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.38e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.50e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.06e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 527.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.20e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 563.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.72e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.15e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.99e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.84e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.37e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.56e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.99e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.77e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.61e3T + 8.58e9T^{2} \) |
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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
−9.792051134227166344787243498883, −8.918214499733086996448889687510, −7.59860791640419179543533365800, −6.60786433281029563974023444411, −5.71494061758803687755186073687, −5.02810619543587909995793617996, −3.91023240503214108497189025269, −2.98152379677419282895592800830, −1.87885092642258534482587555245, 0,
1.87885092642258534482587555245, 2.98152379677419282895592800830, 3.91023240503214108497189025269, 5.02810619543587909995793617996, 5.71494061758803687755186073687, 6.60786433281029563974023444411, 7.59860791640419179543533365800, 8.918214499733086996448889687510, 9.792051134227166344787243498883
