| L(s) = 1 | − 3-s − 5-s + 4·7-s − 2·9-s − 4·13-s + 15-s + 4·17-s + 4·19-s − 4·21-s − 3·23-s − 4·25-s + 5·27-s − 8·29-s + 9·31-s − 4·35-s + 5·37-s + 4·39-s − 12·41-s − 8·43-s + 2·45-s + 4·47-s + 9·49-s − 4·51-s + 10·53-s − 4·57-s − 7·59-s + 8·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s − 2/3·9-s − 1.10·13-s + 0.258·15-s + 0.970·17-s + 0.917·19-s − 0.872·21-s − 0.625·23-s − 4/5·25-s + 0.962·27-s − 1.48·29-s + 1.61·31-s − 0.676·35-s + 0.821·37-s + 0.640·39-s − 1.87·41-s − 1.21·43-s + 0.298·45-s + 0.583·47-s + 9/7·49-s − 0.560·51-s + 1.37·53-s − 0.529·57-s − 0.911·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.71279125859830793317848648239, −6.93911760769520774806312020830, −5.91038766480902031773540339823, −5.32810611728728627420216178235, −4.88723530710282184662191412185, −4.06031099381135244704561271422, −3.11829082930502406226123905947, −2.16255454555790038399051937173, −1.19623289269108085493293258304, 0,
1.19623289269108085493293258304, 2.16255454555790038399051937173, 3.11829082930502406226123905947, 4.06031099381135244704561271422, 4.88723530710282184662191412185, 5.32810611728728627420216178235, 5.91038766480902031773540339823, 6.93911760769520774806312020830, 7.71279125859830793317848648239
