| L(s) = 1 | + 3.24·3-s + 0.109·5-s − 3.44·7-s + 7.54·9-s + 1.13·11-s − 1.85·13-s + 0.356·15-s − 6.65·17-s − 1.93·19-s − 11.1·21-s + 2.40·23-s − 4.98·25-s + 14.7·27-s − 5.96·29-s − 10.7·31-s + 3.69·33-s − 0.378·35-s + 8.67·37-s − 6.00·39-s − 4.29·41-s − 4.18·43-s + 0.829·45-s + 2.63·47-s + 4.86·49-s − 21.6·51-s − 0.472·53-s + 0.124·55-s + ⋯ |
| L(s) = 1 | + 1.87·3-s + 0.0491·5-s − 1.30·7-s + 2.51·9-s + 0.342·11-s − 0.513·13-s + 0.0921·15-s − 1.61·17-s − 0.444·19-s − 2.44·21-s + 0.501·23-s − 0.997·25-s + 2.83·27-s − 1.10·29-s − 1.93·31-s + 0.642·33-s − 0.0640·35-s + 1.42·37-s − 0.962·39-s − 0.670·41-s − 0.638·43-s + 0.123·45-s + 0.383·47-s + 0.695·49-s − 3.02·51-s − 0.0648·53-s + 0.0168·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 113 | \( 1 + T \) |
| good | 3 | \( 1 - 3.24T + 3T^{2} \) |
| 5 | \( 1 - 0.109T + 5T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 - 1.13T + 11T^{2} \) |
| 13 | \( 1 + 1.85T + 13T^{2} \) |
| 17 | \( 1 + 6.65T + 17T^{2} \) |
| 19 | \( 1 + 1.93T + 19T^{2} \) |
| 23 | \( 1 - 2.40T + 23T^{2} \) |
| 29 | \( 1 + 5.96T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 - 8.67T + 37T^{2} \) |
| 41 | \( 1 + 4.29T + 41T^{2} \) |
| 43 | \( 1 + 4.18T + 43T^{2} \) |
| 47 | \( 1 - 2.63T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 + 6.02T + 59T^{2} \) |
| 61 | \( 1 - 5.64T + 61T^{2} \) |
| 67 | \( 1 - 2.51T + 67T^{2} \) |
| 71 | \( 1 - 8.27T + 71T^{2} \) |
| 73 | \( 1 - 1.87T + 73T^{2} \) |
| 79 | \( 1 + 2.22T + 79T^{2} \) |
| 83 | \( 1 + 3.19T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 7.10T + 97T^{2} \) |
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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
−7.63320568288329375281410526742, −6.92659175696632362286711609335, −6.55959933129827714260756108095, −5.45017788822583002272961828594, −4.24568557181065151461588792422, −3.87395073392985375073540745691, −3.09296170540292228191550077885, −2.37030281773904766517492773670, −1.73910437331466841035033577629, 0,
1.73910437331466841035033577629, 2.37030281773904766517492773670, 3.09296170540292228191550077885, 3.87395073392985375073540745691, 4.24568557181065151461588792422, 5.45017788822583002272961828594, 6.55959933129827714260756108095, 6.92659175696632362286711609335, 7.63320568288329375281410526742
