| L(s) = 1 | + 0.732·2-s + 3-s − 1.46·4-s − 3.73·5-s + 0.732·6-s − 7-s − 2.53·8-s + 9-s − 2.73·10-s − 5·11-s − 1.46·12-s + 3.19·13-s − 0.732·14-s − 3.73·15-s + 1.07·16-s + 17-s + 0.732·18-s − 7.19·19-s + 5.46·20-s − 21-s − 3.66·22-s − 3·23-s − 2.53·24-s + 8.92·25-s + 2.33·26-s + 27-s + 1.46·28-s + ⋯ |
| L(s) = 1 | + 0.517·2-s + 0.577·3-s − 0.732·4-s − 1.66·5-s + 0.298·6-s − 0.377·7-s − 0.896·8-s + 0.333·9-s − 0.863·10-s − 1.50·11-s − 0.422·12-s + 0.886·13-s − 0.195·14-s − 0.963·15-s + 0.267·16-s + 0.242·17-s + 0.172·18-s − 1.65·19-s + 1.22·20-s − 0.218·21-s − 0.780·22-s − 0.625·23-s − 0.517·24-s + 1.78·25-s + 0.458·26-s + 0.192·27-s + 0.276·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 0.732T + 2T^{2} \) |
| 5 | \( 1 + 3.73T + 5T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 - 3.19T + 13T^{2} \) |
| 19 | \( 1 + 7.19T + 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 4.73T + 31T^{2} \) |
| 37 | \( 1 - 4.73T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 9.39T + 43T^{2} \) |
| 47 | \( 1 - 4.73T + 47T^{2} \) |
| 53 | \( 1 + 5.66T + 53T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 + 6.73T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 + 3.46T + 73T^{2} \) |
| 79 | \( 1 + 0.196T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−11.01453987916948577263107904695, −10.18102636782805191840638328072, −8.761652782659469796278756010595, −8.294076737002935542744882379183, −7.44656816178385339730106048317, −6.01365118178431469274790105517, −4.62576104131150402797670106134, −3.89192594289745580346062293720, −2.94235397167953120876604826438, 0,
2.94235397167953120876604826438, 3.89192594289745580346062293720, 4.62576104131150402797670106134, 6.01365118178431469274790105517, 7.44656816178385339730106048317, 8.294076737002935542744882379183, 8.761652782659469796278756010595, 10.18102636782805191840638328072, 11.01453987916948577263107904695
