| L(s) = 1 | + 0.570·3-s + 5-s − 7-s − 2.67·9-s + 2.87·11-s − 4.91·13-s + 0.570·15-s − 17-s − 4.87·19-s − 0.570·21-s + 7.08·23-s + 25-s − 3.23·27-s + 3.77·29-s − 5.40·31-s + 1.63·33-s − 35-s − 2.57·37-s − 2.79·39-s − 12.2·41-s − 0.229·43-s − 2.67·45-s − 13.6·47-s + 49-s − 0.570·51-s + 0.340·53-s + 2.87·55-s + ⋯ |
| L(s) = 1 | + 0.329·3-s + 0.447·5-s − 0.377·7-s − 0.891·9-s + 0.866·11-s − 1.36·13-s + 0.147·15-s − 0.242·17-s − 1.11·19-s − 0.124·21-s + 1.47·23-s + 0.200·25-s − 0.622·27-s + 0.700·29-s − 0.971·31-s + 0.285·33-s − 0.169·35-s − 0.422·37-s − 0.448·39-s − 1.91·41-s − 0.0350·43-s − 0.398·45-s − 1.99·47-s + 0.142·49-s − 0.0798·51-s + 0.0467·53-s + 0.387·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 0.570T + 3T^{2} \) |
| 11 | \( 1 - 2.87T + 11T^{2} \) |
| 13 | \( 1 + 4.91T + 13T^{2} \) |
| 19 | \( 1 + 4.87T + 19T^{2} \) |
| 23 | \( 1 - 7.08T + 23T^{2} \) |
| 29 | \( 1 - 3.77T + 29T^{2} \) |
| 31 | \( 1 + 5.40T + 31T^{2} \) |
| 37 | \( 1 + 2.57T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 0.229T + 43T^{2} \) |
| 47 | \( 1 + 13.6T + 47T^{2} \) |
| 53 | \( 1 - 0.340T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 3.44T + 71T^{2} \) |
| 73 | \( 1 + 3.20T + 73T^{2} \) |
| 79 | \( 1 - 2.49T + 79T^{2} \) |
| 83 | \( 1 + 0.463T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 + 7.88T + 97T^{2} \) |
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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
−8.850317907764668335185182387053, −7.895466727588310164997860421144, −6.84061663450190884072469206016, −6.47427285062738810647947198113, −5.35479290880114671300110412150, −4.69436131828863439068378798639, −3.48937019693123790897428300287, −2.72257116209349534779150935999, −1.74113304174251992435324191213, 0,
1.74113304174251992435324191213, 2.72257116209349534779150935999, 3.48937019693123790897428300287, 4.69436131828863439068378798639, 5.35479290880114671300110412150, 6.47427285062738810647947198113, 6.84061663450190884072469206016, 7.895466727588310164997860421144, 8.850317907764668335185182387053
