| L(s) = 1 | − 0.336·2-s + 3-s − 1.88·4-s − 0.687·5-s − 0.336·6-s − 7-s + 1.30·8-s + 9-s + 0.231·10-s + 1.68·11-s − 1.88·12-s − 6.81·13-s + 0.336·14-s − 0.687·15-s + 3.33·16-s + 6.85·17-s − 0.336·18-s + 2.14·19-s + 1.29·20-s − 21-s − 0.568·22-s − 0.0626·23-s + 1.30·24-s − 4.52·25-s + 2.29·26-s + 27-s + 1.88·28-s + ⋯ |
| L(s) = 1 | − 0.238·2-s + 0.577·3-s − 0.943·4-s − 0.307·5-s − 0.137·6-s − 0.377·7-s + 0.462·8-s + 0.333·9-s + 0.0731·10-s + 0.509·11-s − 0.544·12-s − 1.89·13-s + 0.0899·14-s − 0.177·15-s + 0.833·16-s + 1.66·17-s − 0.0793·18-s + 0.493·19-s + 0.289·20-s − 0.218·21-s − 0.121·22-s − 0.0130·23-s + 0.267·24-s − 0.905·25-s + 0.450·26-s + 0.192·27-s + 0.356·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 0.336T + 2T^{2} \) |
| 5 | \( 1 + 0.687T + 5T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 + 6.81T + 13T^{2} \) |
| 17 | \( 1 - 6.85T + 17T^{2} \) |
| 19 | \( 1 - 2.14T + 19T^{2} \) |
| 23 | \( 1 + 0.0626T + 23T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 31 | \( 1 + 8.26T + 31T^{2} \) |
| 37 | \( 1 - 7.09T + 37T^{2} \) |
| 41 | \( 1 - 2.42T + 41T^{2} \) |
| 43 | \( 1 - 7.67T + 43T^{2} \) |
| 47 | \( 1 + 5.80T + 47T^{2} \) |
| 53 | \( 1 - 3.43T + 53T^{2} \) |
| 59 | \( 1 - 6.87T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 0.658T + 73T^{2} \) |
| 79 | \( 1 + 3.47T + 79T^{2} \) |
| 83 | \( 1 + 3.30T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−7.65207123287116044907713072554, −7.20906611372874867077975784669, −5.96521629223945788665437276445, −5.33414533323650533339875529425, −4.58205223642381022442667460877, −3.82141156038196435149653285753, −3.25743538443317068108639155710, −2.26317979557715373364130706301, −1.12433425442133628489646536751, 0,
1.12433425442133628489646536751, 2.26317979557715373364130706301, 3.25743538443317068108639155710, 3.82141156038196435149653285753, 4.58205223642381022442667460877, 5.33414533323650533339875529425, 5.96521629223945788665437276445, 7.20906611372874867077975784669, 7.65207123287116044907713072554
