| L(s) = 1 | − 3-s + 5-s − 2·9-s + 5·11-s − 13-s − 15-s − 6·17-s − 4·19-s − 6·23-s − 4·25-s + 5·27-s + 29-s + 9·31-s − 5·33-s + 39-s − 8·41-s + 43-s − 2·45-s + 9·47-s − 7·49-s + 6·51-s + 9·53-s + 5·55-s + 4·57-s − 14·59-s − 10·61-s − 65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 2/3·9-s + 1.50·11-s − 0.277·13-s − 0.258·15-s − 1.45·17-s − 0.917·19-s − 1.25·23-s − 4/5·25-s + 0.962·27-s + 0.185·29-s + 1.61·31-s − 0.870·33-s + 0.160·39-s − 1.24·41-s + 0.152·43-s − 0.298·45-s + 1.31·47-s − 49-s + 0.840·51-s + 1.23·53-s + 0.674·55-s + 0.529·57-s − 1.82·59-s − 1.28·61-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−8.821811966967850494081800766308, −8.281308926670353277494816830699, −7.00188035007504764283689529378, −6.26094632564158582902254335626, −5.94099511737124794616567757823, −4.65525561772861292990274176806, −4.05445855657433236191809071172, −2.67613733139012553443753048260, −1.63714266819942374366659837510, 0,
1.63714266819942374366659837510, 2.67613733139012553443753048260, 4.05445855657433236191809071172, 4.65525561772861292990274176806, 5.94099511737124794616567757823, 6.26094632564158582902254335626, 7.00188035007504764283689529378, 8.281308926670353277494816830699, 8.821811966967850494081800766308
