| L(s) = 1 | − 3-s + 9-s − 4·13-s + 3·17-s − 2·19-s − 3·23-s − 27-s + 31-s − 10·37-s + 4·39-s − 9·41-s + 10·43-s + 3·47-s − 3·51-s − 6·53-s + 2·57-s + 6·59-s + 8·61-s + 4·67-s + 3·69-s − 3·71-s + 14·73-s − 11·79-s + 81-s − 15·89-s − 93-s − 7·97-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.10·13-s + 0.727·17-s − 0.458·19-s − 0.625·23-s − 0.192·27-s + 0.179·31-s − 1.64·37-s + 0.640·39-s − 1.40·41-s + 1.52·43-s + 0.437·47-s − 0.420·51-s − 0.824·53-s + 0.264·57-s + 0.781·59-s + 1.02·61-s + 0.488·67-s + 0.361·69-s − 0.356·71-s + 1.63·73-s − 1.23·79-s + 1/9·81-s − 1.58·89-s − 0.103·93-s − 0.710·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.49402806003368, −14.09194549741234, −13.71645884076788, −12.77700665785956, −12.57402955742020, −12.12214645711111, −11.55751362146952, −11.11481703628134, −10.37465737551528, −10.05082297267882, −9.670821176768727, −8.872126992308200, −8.387263604657127, −7.761810364624561, −7.156145182864108, −6.826311445283575, −6.063455145734961, −5.541043638155508, −5.046674837552617, −4.471599975080035, −3.795535928219144, −3.169223861596981, −2.312574095264917, −1.774280168302786, −0.8017811877111340, 0,
0.8017811877111340, 1.774280168302786, 2.312574095264917, 3.169223861596981, 3.795535928219144, 4.471599975080035, 5.046674837552617, 5.541043638155508, 6.063455145734961, 6.826311445283575, 7.156145182864108, 7.761810364624561, 8.387263604657127, 8.872126992308200, 9.670821176768727, 10.05082297267882, 10.37465737551528, 11.11481703628134, 11.55751362146952, 12.12214645711111, 12.57402955742020, 12.77700665785956, 13.71645884076788, 14.09194549741234, 14.49402806003368
