| L(s) = 1 | − 3-s + 9-s − 2·11-s − 2·13-s + 4·17-s + 8·23-s − 27-s + 2·31-s + 2·33-s − 8·37-s + 2·39-s − 2·41-s − 2·43-s + 10·47-s − 4·51-s + 2·53-s − 4·59-s − 10·61-s + 2·67-s − 8·69-s + 12·71-s − 10·73-s − 16·79-s + 81-s + 16·83-s + 14·89-s − 2·93-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.970·17-s + 1.66·23-s − 0.192·27-s + 0.359·31-s + 0.348·33-s − 1.31·37-s + 0.320·39-s − 0.312·41-s − 0.304·43-s + 1.45·47-s − 0.560·51-s + 0.274·53-s − 0.520·59-s − 1.28·61-s + 0.244·67-s − 0.963·69-s + 1.42·71-s − 1.17·73-s − 1.80·79-s + 1/9·81-s + 1.75·83-s + 1.48·89-s − 0.207·93-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 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.67193731325115, −14.00313469928911, −13.57664754420637, −13.00370239432988, −12.46898049493291, −12.09949500031759, −11.64231294208160, −10.87641296744209, −10.58968047641242, −10.11373552866121, −9.483195807162827, −8.977358265092181, −8.374313049178115, −7.666146769569592, −7.298366521545048, −6.754573381142410, −6.118523031176463, −5.370376761131070, −5.158131423219626, −4.539555614333121, −3.740292549243682, −3.073282918140951, −2.524857016771351, −1.586271667304367, −0.8950729294548966, 0,
0.8950729294548966, 1.586271667304367, 2.524857016771351, 3.073282918140951, 3.740292549243682, 4.539555614333121, 5.158131423219626, 5.370376761131070, 6.118523031176463, 6.754573381142410, 7.298366521545048, 7.666146769569592, 8.374313049178115, 8.977358265092181, 9.483195807162827, 10.11373552866121, 10.58968047641242, 10.87641296744209, 11.64231294208160, 12.09949500031759, 12.46898049493291, 13.00370239432988, 13.57664754420637, 14.00313469928911, 14.67193731325115
