L(s) = 1 | − 2-s + 0.732·3-s + 4-s − 1.73·5-s − 0.732·6-s − 4.73·7-s − 8-s − 2.46·9-s + 1.73·10-s + 0.732·12-s − 3·13-s + 4.73·14-s − 1.26·15-s + 16-s + 5.19·17-s + 2.46·18-s − 1.26·19-s − 1.73·20-s − 3.46·21-s − 1.26·23-s − 0.732·24-s − 2.00·25-s + 3·26-s − 4·27-s − 4.73·28-s + 3·29-s + 1.26·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.422·3-s + 0.5·4-s − 0.774·5-s − 0.298·6-s − 1.78·7-s − 0.353·8-s − 0.821·9-s + 0.547·10-s + 0.211·12-s − 0.832·13-s + 1.26·14-s − 0.327·15-s + 0.250·16-s + 1.26·17-s + 0.580·18-s − 0.290·19-s − 0.387·20-s − 0.755·21-s − 0.264·23-s − 0.149·24-s − 0.400·25-s + 0.588·26-s − 0.769·27-s − 0.894·28-s + 0.557·29-s + 0.231·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 + 1.26T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 0.196T + 31T^{2} \) |
| 37 | \( 1 + 7.19T + 37T^{2} \) |
| 41 | \( 1 - 0.803T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 8.19T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 9.46T + 59T^{2} \) |
| 61 | \( 1 + 2.53T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 - 0.928T + 73T^{2} \) |
| 79 | \( 1 - 4.73T + 79T^{2} \) |
| 83 | \( 1 + 8.19T + 83T^{2} \) |
| 89 | \( 1 - 6.46T + 89T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−11.79603484416643859257716409015, −10.35281506916873688680540010436, −9.699921877978840987783112143188, −8.743633813748813002004319794376, −7.78156545562129664703301943240, −6.83052973402609462766702911524, −5.68136949190278382121457579315, −3.66771694832268838501720764616, −2.77244735190835634803773504009, 0,
2.77244735190835634803773504009, 3.66771694832268838501720764616, 5.68136949190278382121457579315, 6.83052973402609462766702911524, 7.78156545562129664703301943240, 8.743633813748813002004319794376, 9.699921877978840987783112143188, 10.35281506916873688680540010436, 11.79603484416643859257716409015