| L(s) = 1 | + 5·5-s + 6·7-s − 60·11-s + 50·13-s + 30·17-s + 40·19-s − 178·23-s + 25·25-s − 166·29-s + 20·31-s + 30·35-s + 10·37-s + 250·41-s + 142·43-s − 214·47-s − 307·49-s − 490·53-s − 300·55-s + 800·59-s + 250·61-s + 250·65-s − 774·67-s − 100·71-s − 230·73-s − 360·77-s − 1.32e3·79-s − 982·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.323·7-s − 1.64·11-s + 1.06·13-s + 0.428·17-s + 0.482·19-s − 1.61·23-s + 1/5·25-s − 1.06·29-s + 0.115·31-s + 0.144·35-s + 0.0444·37-s + 0.952·41-s + 0.503·43-s − 0.664·47-s − 0.895·49-s − 1.26·53-s − 0.735·55-s + 1.76·59-s + 0.524·61-s + 0.477·65-s − 1.41·67-s − 0.167·71-s − 0.368·73-s − 0.532·77-s − 1.87·79-s − 1.29·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| good | 7 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 50 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 40 T + p^{3} T^{2} \) |
| 23 | \( 1 + 178 T + p^{3} T^{2} \) |
| 29 | \( 1 + 166 T + p^{3} T^{2} \) |
| 31 | \( 1 - 20 T + p^{3} T^{2} \) |
| 37 | \( 1 - 10 T + p^{3} T^{2} \) |
| 41 | \( 1 - 250 T + p^{3} T^{2} \) |
| 43 | \( 1 - 142 T + p^{3} T^{2} \) |
| 47 | \( 1 + 214 T + p^{3} T^{2} \) |
| 53 | \( 1 + 490 T + p^{3} T^{2} \) |
| 59 | \( 1 - 800 T + p^{3} T^{2} \) |
| 61 | \( 1 - 250 T + p^{3} T^{2} \) |
| 67 | \( 1 + 774 T + p^{3} T^{2} \) |
| 71 | \( 1 + 100 T + p^{3} T^{2} \) |
| 73 | \( 1 + 230 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1320 T + p^{3} T^{2} \) |
| 83 | \( 1 + 982 T + p^{3} T^{2} \) |
| 89 | \( 1 + 874 T + p^{3} T^{2} \) |
| 97 | \( 1 + 310 T + p^{3} T^{2} \) |
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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
−8.631110007440500643643641929041, −7.975291981124062710162686327239, −7.32040008714933576267079431405, −5.99884457062361662239430896150, −5.62401479097580832018562461052, −4.59655277249171470934803855051, −3.49836051256242586291057072221, −2.46129211346905869168313225818, −1.43201257363883792789469627823, 0,
1.43201257363883792789469627823, 2.46129211346905869168313225818, 3.49836051256242586291057072221, 4.59655277249171470934803855051, 5.62401479097580832018562461052, 5.99884457062361662239430896150, 7.32040008714933576267079431405, 7.975291981124062710162686327239, 8.631110007440500643643641929041
