| L(s) = 1 | − 5-s + 2·7-s + 13-s + 23-s + 25-s + 3·29-s − 3·31-s − 2·35-s − 8·37-s − 3·41-s + 2·43-s − 11·47-s − 3·49-s + 14·53-s − 8·59-s − 4·61-s − 65-s + 4·67-s + 7·71-s − 9·73-s + 4·83-s + 2·89-s + 2·91-s + 18·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 0.277·13-s + 0.208·23-s + 1/5·25-s + 0.557·29-s − 0.538·31-s − 0.338·35-s − 1.31·37-s − 0.468·41-s + 0.304·43-s − 1.60·47-s − 3/7·49-s + 1.92·53-s − 1.04·59-s − 0.512·61-s − 0.124·65-s + 0.488·67-s + 0.830·71-s − 1.05·73-s + 0.439·83-s + 0.211·89-s + 0.209·91-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−16.07141904797211, −15.72380292527337, −14.96794610955298, −14.67085612758663, −14.01886715133039, −13.45155038191190, −12.90526581641141, −12.10909449493302, −11.83207076854151, −11.13764383361161, −10.66013821826660, −10.11763170804765, −9.273624840080437, −8.749028320623113, −8.137797932049989, −7.704260315207067, −6.907313159197437, −6.449308649692143, −5.457132014078334, −5.039325556459056, −4.288071147324384, −3.620971522245182, −2.891708127395845, −1.918058480818782, −1.190758614153488, 0,
1.190758614153488, 1.918058480818782, 2.891708127395845, 3.620971522245182, 4.288071147324384, 5.039325556459056, 5.457132014078334, 6.449308649692143, 6.907313159197437, 7.704260315207067, 8.137797932049989, 8.749028320623113, 9.273624840080437, 10.11763170804765, 10.66013821826660, 11.13764383361161, 11.83207076854151, 12.10909449493302, 12.90526581641141, 13.45155038191190, 14.01886715133039, 14.67085612758663, 14.96794610955298, 15.72380292527337, 16.07141904797211
