| L(s) = 1 | − 11-s + 2·13-s + 6·17-s − 4·23-s + 2·29-s − 10·37-s − 6·41-s + 8·43-s + 4·47-s − 7·49-s + 6·53-s + 12·59-s − 2·61-s − 4·67-s + 12·71-s + 14·73-s − 16·79-s − 12·83-s − 10·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.301·11-s + 0.554·13-s + 1.45·17-s − 0.834·23-s + 0.371·29-s − 1.64·37-s − 0.937·41-s + 1.21·43-s + 0.583·47-s − 49-s + 0.824·53-s + 1.56·59-s − 0.256·61-s − 0.488·67-s + 1.42·71-s + 1.63·73-s − 1.80·79-s − 1.31·83-s − 1.05·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.72121943524729, −12.85305475784576, −12.66880809504147, −12.10887207994906, −11.63697611517644, −11.24670154343183, −10.49635078516983, −10.21838498096174, −9.856595504156537, −9.194551327487356, −8.602853470397764, −8.272390425814639, −7.740576562859163, −7.222086625716444, −6.708181048028999, −6.124255867034991, −5.499384744877471, −5.309225192557394, −4.542865003613828, −3.812711412180327, −3.546523362421847, −2.834479270763527, −2.200184740863032, −1.486236385012563, −0.8914759082094120, 0,
0.8914759082094120, 1.486236385012563, 2.200184740863032, 2.834479270763527, 3.546523362421847, 3.812711412180327, 4.542865003613828, 5.309225192557394, 5.499384744877471, 6.124255867034991, 6.708181048028999, 7.222086625716444, 7.740576562859163, 8.272390425814639, 8.602853470397764, 9.194551327487356, 9.856595504156537, 10.21838498096174, 10.49635078516983, 11.24670154343183, 11.63697611517644, 12.10887207994906, 12.66880809504147, 12.85305475784576, 13.72121943524729
