| L(s) = 1 | + 0.267·2-s + 2.13·3-s − 1.92·4-s − 0.961·5-s + 0.570·6-s − 2.53·7-s − 1.04·8-s + 1.55·9-s − 0.256·10-s + 4.97·11-s − 4.11·12-s − 13-s − 0.677·14-s − 2.05·15-s + 3.57·16-s − 2.24·17-s + 0.416·18-s − 6.79·19-s + 1.85·20-s − 5.41·21-s + 1.32·22-s − 1.51·23-s − 2.24·24-s − 4.07·25-s − 0.267·26-s − 3.07·27-s + 4.88·28-s + ⋯ |
| L(s) = 1 | + 0.188·2-s + 1.23·3-s − 0.964·4-s − 0.429·5-s + 0.232·6-s − 0.958·7-s − 0.371·8-s + 0.519·9-s − 0.0812·10-s + 1.49·11-s − 1.18·12-s − 0.277·13-s − 0.181·14-s − 0.529·15-s + 0.894·16-s − 0.544·17-s + 0.0981·18-s − 1.55·19-s + 0.414·20-s − 1.18·21-s + 0.283·22-s − 0.314·23-s − 0.457·24-s − 0.815·25-s − 0.0524·26-s − 0.592·27-s + 0.924·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 0.267T + 2T^{2} \) |
| 3 | \( 1 - 2.13T + 3T^{2} \) |
| 5 | \( 1 + 0.961T + 5T^{2} \) |
| 7 | \( 1 + 2.53T + 7T^{2} \) |
| 11 | \( 1 - 4.97T + 11T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 + 6.79T + 19T^{2} \) |
| 23 | \( 1 + 1.51T + 23T^{2} \) |
| 29 | \( 1 - 0.689T + 29T^{2} \) |
| 31 | \( 1 + 2.89T + 31T^{2} \) |
| 37 | \( 1 + 8.06T + 37T^{2} \) |
| 41 | \( 1 - 3.66T + 41T^{2} \) |
| 43 | \( 1 + 2.86T + 43T^{2} \) |
| 47 | \( 1 - 0.642T + 47T^{2} \) |
| 53 | \( 1 + 8.05T + 53T^{2} \) |
| 59 | \( 1 - 1.97T + 59T^{2} \) |
| 61 | \( 1 + 7.25T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 3.67T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 4.84T + 79T^{2} \) |
| 83 | \( 1 + 1.47T + 83T^{2} \) |
| 89 | \( 1 - 9.83T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−9.176703206454353079879815149311, −8.590601180991502437760557572380, −7.84744527966166305415275339718, −6.74078093338890415863369737115, −6.01030114417538073996366173622, −4.56529759836697609077125174967, −3.82974479836200085364994577289, −3.32377076392852673727922287630, −1.97270379070600412409044583764, 0,
1.97270379070600412409044583764, 3.32377076392852673727922287630, 3.82974479836200085364994577289, 4.56529759836697609077125174967, 6.01030114417538073996366173622, 6.74078093338890415863369737115, 7.84744527966166305415275339718, 8.590601180991502437760557572380, 9.176703206454353079879815149311
