| L(s) = 1 | − 2.31·2-s − 2.05·3-s + 3.37·4-s + 5-s + 4.75·6-s + 0.529·7-s − 3.20·8-s + 1.20·9-s − 2.31·10-s − 1.91·11-s − 6.92·12-s − 3.07·13-s − 1.22·14-s − 2.05·15-s + 0.663·16-s − 17-s − 2.79·18-s − 3.37·19-s + 3.37·20-s − 1.08·21-s + 4.44·22-s + 0.777·23-s + 6.56·24-s + 25-s + 7.14·26-s + 3.68·27-s + 1.79·28-s + ⋯ |
| L(s) = 1 | − 1.64·2-s − 1.18·3-s + 1.68·4-s + 0.447·5-s + 1.94·6-s + 0.200·7-s − 1.13·8-s + 0.400·9-s − 0.733·10-s − 0.577·11-s − 2.00·12-s − 0.853·13-s − 0.328·14-s − 0.529·15-s + 0.165·16-s − 0.242·17-s − 0.657·18-s − 0.774·19-s + 0.755·20-s − 0.236·21-s + 0.947·22-s + 0.162·23-s + 1.33·24-s + 0.200·25-s + 1.40·26-s + 0.709·27-s + 0.338·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 + 2.31T + 2T^{2} \) |
| 3 | \( 1 + 2.05T + 3T^{2} \) |
| 7 | \( 1 - 0.529T + 7T^{2} \) |
| 11 | \( 1 + 1.91T + 11T^{2} \) |
| 13 | \( 1 + 3.07T + 13T^{2} \) |
| 19 | \( 1 + 3.37T + 19T^{2} \) |
| 23 | \( 1 - 0.777T + 23T^{2} \) |
| 29 | \( 1 + 9.08T + 29T^{2} \) |
| 31 | \( 1 + 0.124T + 31T^{2} \) |
| 37 | \( 1 - 8.52T + 37T^{2} \) |
| 41 | \( 1 - 9.07T + 41T^{2} \) |
| 43 | \( 1 - 7.55T + 43T^{2} \) |
| 47 | \( 1 - 2.06T + 47T^{2} \) |
| 53 | \( 1 - 3.24T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 0.275T + 61T^{2} \) |
| 67 | \( 1 + 7.61T + 67T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 1.22T + 79T^{2} \) |
| 83 | \( 1 - 2.37T + 83T^{2} \) |
| 89 | \( 1 - 5.54T + 89T^{2} \) |
| 97 | \( 1 + 0.175T + 97T^{2} \) |
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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
−7.68316561272907560705048059948, −7.24213727093150783599238831675, −6.36233631934839690209777034381, −5.84373980417999164817547048758, −5.05409814736963213817563293413, −4.24874662313630591831269102107, −2.66490120514419189530990461334, −2.04545556738605690648274025901, −0.883848560745588552723914644980, 0,
0.883848560745588552723914644980, 2.04545556738605690648274025901, 2.66490120514419189530990461334, 4.24874662313630591831269102107, 5.05409814736963213817563293413, 5.84373980417999164817547048758, 6.36233631934839690209777034381, 7.24213727093150783599238831675, 7.68316561272907560705048059948
