| L(s) = 1 | − 3-s − 3·7-s + 9-s + 11-s − 13-s − 6·19-s + 3·21-s + 23-s − 27-s − 4·29-s + 3·31-s − 33-s − 5·37-s + 39-s − 11·41-s − 4·43-s + 12·47-s + 2·49-s + 6·57-s − 9·59-s − 4·61-s − 3·63-s − 13·67-s − 69-s − 2·71-s − 16·73-s − 3·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 1.37·19-s + 0.654·21-s + 0.208·23-s − 0.192·27-s − 0.742·29-s + 0.538·31-s − 0.174·33-s − 0.821·37-s + 0.160·39-s − 1.71·41-s − 0.609·43-s + 1.75·47-s + 2/7·49-s + 0.794·57-s − 1.17·59-s − 0.512·61-s − 0.377·63-s − 1.58·67-s − 0.120·69-s − 0.237·71-s − 1.87·73-s − 0.341·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.79886811095416, −12.48082289180721, −11.96590285203657, −11.68863986946616, −11.03220947191607, −10.49152645702410, −10.26648247766989, −9.823556114036212, −9.213542425441272, −8.760698917794743, −8.506855987667624, −7.568915236245131, −7.299775124806670, −6.721593264896021, −6.330299346160701, −5.955995749417247, −5.441481243867433, −4.745853952363518, −4.366691547808816, −3.789786619742447, −3.189156514336685, −2.787021860843181, −1.903768495393407, −1.532287269581122, −0.5092432544613200, 0,
0.5092432544613200, 1.532287269581122, 1.903768495393407, 2.787021860843181, 3.189156514336685, 3.789786619742447, 4.366691547808816, 4.745853952363518, 5.441481243867433, 5.955995749417247, 6.330299346160701, 6.721593264896021, 7.299775124806670, 7.568915236245131, 8.506855987667624, 8.760698917794743, 9.213542425441272, 9.823556114036212, 10.26648247766989, 10.49152645702410, 11.03220947191607, 11.68863986946616, 11.96590285203657, 12.48082289180721, 12.79886811095416
