| L(s) = 1 | − 25·5-s − 32·7-s + 12·11-s − 154·13-s + 918·17-s + 1.06e3·19-s − 4.22e3·23-s + 625·25-s + 7.89e3·29-s − 5.19e3·31-s + 800·35-s + 1.63e4·37-s − 3.64e3·41-s − 1.51e4·43-s + 2.35e4·47-s − 1.57e4·49-s + 1.60e4·53-s − 300·55-s − 1.43e4·59-s − 4.79e4·61-s + 3.85e3·65-s − 3.30e4·67-s + 5.19e4·71-s + 1.20e4·73-s − 384·77-s − 2.51e4·79-s + 3.57e4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.246·7-s + 0.0299·11-s − 0.252·13-s + 0.770·17-s + 0.673·19-s − 1.66·23-s + 1/5·25-s + 1.74·29-s − 0.970·31-s + 0.110·35-s + 1.96·37-s − 0.338·41-s − 1.24·43-s + 1.55·47-s − 0.939·49-s + 0.786·53-s − 0.0133·55-s − 0.536·59-s − 1.64·61-s + 0.113·65-s − 0.900·67-s + 1.22·71-s + 0.264·73-s − 0.00738·77-s − 0.453·79-s + 0.570·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + p^{2} T \) |
| good | 7 | \( 1 + 32 T + p^{5} T^{2} \) |
| 11 | \( 1 - 12 T + p^{5} T^{2} \) |
| 13 | \( 1 + 154 T + p^{5} T^{2} \) |
| 17 | \( 1 - 54 p T + p^{5} T^{2} \) |
| 19 | \( 1 - 1060 T + p^{5} T^{2} \) |
| 23 | \( 1 + 4224 T + p^{5} T^{2} \) |
| 29 | \( 1 - 7890 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5192 T + p^{5} T^{2} \) |
| 37 | \( 1 - 16382 T + p^{5} T^{2} \) |
| 41 | \( 1 + 3642 T + p^{5} T^{2} \) |
| 43 | \( 1 + 15116 T + p^{5} T^{2} \) |
| 47 | \( 1 - 23592 T + p^{5} T^{2} \) |
| 53 | \( 1 - 16074 T + p^{5} T^{2} \) |
| 59 | \( 1 + 14340 T + p^{5} T^{2} \) |
| 61 | \( 1 + 47938 T + p^{5} T^{2} \) |
| 67 | \( 1 + 33092 T + p^{5} T^{2} \) |
| 71 | \( 1 - 51912 T + p^{5} T^{2} \) |
| 73 | \( 1 - 12026 T + p^{5} T^{2} \) |
| 79 | \( 1 + 25160 T + p^{5} T^{2} \) |
| 83 | \( 1 - 35796 T + p^{5} T^{2} \) |
| 89 | \( 1 - 75510 T + p^{5} T^{2} \) |
| 97 | \( 1 + 44158 T + p^{5} T^{2} \) |
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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
−9.332072453169295767523742056403, −8.176049373364383466502187272571, −7.63790449442894271282519333828, −6.56435660858250732365508535071, −5.67387769651884296018054383149, −4.58818618674720315632175576115, −3.62019323753599617230750692057, −2.60613016332891873553279482200, −1.20799293174036270921675806288, 0,
1.20799293174036270921675806288, 2.60613016332891873553279482200, 3.62019323753599617230750692057, 4.58818618674720315632175576115, 5.67387769651884296018054383149, 6.56435660858250732365508535071, 7.63790449442894271282519333828, 8.176049373364383466502187272571, 9.332072453169295767523742056403
