| L(s) = 1 | + 6.51·2-s − 24.1·3-s + 10.4·4-s + 25·5-s − 157.·6-s + 214.·7-s − 140.·8-s + 341.·9-s + 162.·10-s − 121·11-s − 253.·12-s + 302.·13-s + 1.39e3·14-s − 604.·15-s − 1.24e3·16-s − 661.·17-s + 2.22e3·18-s + 361·19-s + 261.·20-s − 5.19e3·21-s − 788.·22-s − 4.68e3·23-s + 3.39e3·24-s + 625·25-s + 1.96e3·26-s − 2.37e3·27-s + 2.24e3·28-s + ⋯ |
| L(s) = 1 | + 1.15·2-s − 1.55·3-s + 0.327·4-s + 0.447·5-s − 1.78·6-s + 1.65·7-s − 0.775·8-s + 1.40·9-s + 0.515·10-s − 0.301·11-s − 0.507·12-s + 0.495·13-s + 1.90·14-s − 0.693·15-s − 1.22·16-s − 0.554·17-s + 1.61·18-s + 0.229·19-s + 0.146·20-s − 2.56·21-s − 0.347·22-s − 1.84·23-s + 1.20·24-s + 0.200·25-s + 0.571·26-s − 0.626·27-s + 0.542·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 6.51T + 32T^{2} \) |
| 3 | \( 1 + 24.1T + 243T^{2} \) |
| 7 | \( 1 - 214.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 302.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 661.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 4.68e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 549.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.53e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.21e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.11e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.31e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.72e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.03e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.88e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 8.77e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.51e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.88e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.08e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.01e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.45e4T + 8.58e9T^{2} \) |
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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
−8.657695670393942657831032258077, −7.82829879264171374649821987212, −6.49527029714457115680908422011, −6.01943082715559048929734190417, −5.13171011330029157011008076924, −4.77843933699308675577009126443, −3.91311517765228856970800982060, −2.30918065207581655094096497005, −1.23808405980010421606409530506, 0,
1.23808405980010421606409530506, 2.30918065207581655094096497005, 3.91311517765228856970800982060, 4.77843933699308675577009126443, 5.13171011330029157011008076924, 6.01943082715559048929734190417, 6.49527029714457115680908422011, 7.82829879264171374649821987212, 8.657695670393942657831032258077
