| L(s) = 1 | − 1.61·2-s − 2.23·3-s + 0.618·4-s − 5-s + 3.61·6-s + 0.236·7-s + 2.23·8-s + 2.00·9-s + 1.61·10-s − 4.23·11-s − 1.38·12-s − 0.381·14-s + 2.23·15-s − 4.85·16-s + 5.47·17-s − 3.23·18-s + 0.236·19-s − 0.618·20-s − 0.527·21-s + 6.85·22-s + 8.23·23-s − 5.00·24-s + 25-s + 2.23·27-s + 0.145·28-s + 1.47·29-s − 3.61·30-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 1.29·3-s + 0.309·4-s − 0.447·5-s + 1.47·6-s + 0.0892·7-s + 0.790·8-s + 0.666·9-s + 0.511·10-s − 1.27·11-s − 0.398·12-s − 0.102·14-s + 0.577·15-s − 1.21·16-s + 1.32·17-s − 0.762·18-s + 0.0541·19-s − 0.138·20-s − 0.115·21-s + 1.46·22-s + 1.71·23-s − 1.02·24-s + 0.200·25-s + 0.430·27-s + 0.0275·28-s + 0.273·29-s − 0.660·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 + 2.23T + 3T^{2} \) |
| 7 | \( 1 - 0.236T + 7T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 17 | \( 1 - 5.47T + 17T^{2} \) |
| 19 | \( 1 - 0.236T + 19T^{2} \) |
| 23 | \( 1 - 8.23T + 23T^{2} \) |
| 29 | \( 1 - 1.47T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 - 5.94T + 41T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 1.76T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 8.94T + 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 + 5.47T + 97T^{2} \) |
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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
−9.923164590698270054817650236949, −8.976502786094128164749537110187, −7.972570524500856780221601387272, −7.46942427363821226489207152276, −6.41775426363009024126125164811, −5.23362312287390411068686025004, −4.76575291969552224028215418120, −3.08380397726926203489760297355, −1.20424520096660075309951790827, 0,
1.20424520096660075309951790827, 3.08380397726926203489760297355, 4.76575291969552224028215418120, 5.23362312287390411068686025004, 6.41775426363009024126125164811, 7.46942427363821226489207152276, 7.972570524500856780221601387272, 8.976502786094128164749537110187, 9.923164590698270054817650236949
