| L(s) = 1 | − 2.61·2-s + 3-s + 4.85·4-s − 3.61·5-s − 2.61·6-s + 7-s − 7.47·8-s + 9-s + 9.47·10-s + 3.47·11-s + 4.85·12-s − 4.61·13-s − 2.61·14-s − 3.61·15-s + 9.85·16-s − 17-s − 2.61·18-s − 3·19-s − 17.5·20-s + 21-s − 9.09·22-s − 23-s − 7.47·24-s + 8.09·25-s + 12.0·26-s + 27-s + 4.85·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.577·3-s + 2.42·4-s − 1.61·5-s − 1.06·6-s + 0.377·7-s − 2.64·8-s + 0.333·9-s + 2.99·10-s + 1.04·11-s + 1.40·12-s − 1.28·13-s − 0.699·14-s − 0.934·15-s + 2.46·16-s − 0.242·17-s − 0.617·18-s − 0.688·19-s − 3.92·20-s + 0.218·21-s − 1.93·22-s − 0.208·23-s − 1.52·24-s + 1.61·25-s + 2.37·26-s + 0.192·27-s + 0.917·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 5 | \( 1 + 3.61T + 5T^{2} \) |
| 11 | \( 1 - 3.47T + 11T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 29 | \( 1 - 1.47T + 29T^{2} \) |
| 31 | \( 1 + 8.23T + 31T^{2} \) |
| 37 | \( 1 + 7.47T + 37T^{2} \) |
| 41 | \( 1 - 8.70T + 41T^{2} \) |
| 43 | \( 1 + 3.09T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 5.61T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 9.47T + 79T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 - 9.85T + 89T^{2} \) |
| 97 | \( 1 - 5T + 97T^{2} \) |
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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
−10.43343526796892694157864761346, −9.327444753066146962134264057681, −8.777510517566518098706293840748, −7.86935605681606903657724655349, −7.42769423515147453862951066724, −6.56845343610710827421950839293, −4.50426503132686146127872626357, −3.24229584203906335485827551561, −1.78219363577744059942077016856, 0,
1.78219363577744059942077016856, 3.24229584203906335485827551561, 4.50426503132686146127872626357, 6.56845343610710827421950839293, 7.42769423515147453862951066724, 7.86935605681606903657724655349, 8.777510517566518098706293840748, 9.327444753066146962134264057681, 10.43343526796892694157864761346
