| L(s) = 1 | + 54·5-s − 49·7-s + 216·11-s + 998·13-s − 1.30e3·17-s − 884·19-s − 2.26e3·23-s − 209·25-s + 1.48e3·29-s − 8.36e3·31-s − 2.64e3·35-s − 4.71e3·37-s + 9.78e3·41-s − 1.94e4·43-s + 2.22e4·47-s + 2.40e3·49-s − 2.67e4·53-s + 1.16e4·55-s + 2.80e4·59-s − 3.88e4·61-s + 5.38e4·65-s − 2.39e4·67-s − 2.06e4·71-s + 290·73-s − 1.05e4·77-s + 9.95e4·79-s + 1.93e4·83-s + ⋯ |
| L(s) = 1 | + 0.965·5-s − 0.377·7-s + 0.538·11-s + 1.63·13-s − 1.09·17-s − 0.561·19-s − 0.893·23-s − 0.0668·25-s + 0.327·29-s − 1.56·31-s − 0.365·35-s − 0.566·37-s + 0.909·41-s − 1.60·43-s + 1.46·47-s + 1/7·49-s − 1.31·53-s + 0.519·55-s + 1.05·59-s − 1.33·61-s + 1.58·65-s − 0.651·67-s − 0.485·71-s + 0.00636·73-s − 0.203·77-s + 1.79·79-s + 0.307·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 54 T + p^{5} T^{2} \) |
| 11 | \( 1 - 216 T + p^{5} T^{2} \) |
| 13 | \( 1 - 998 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1302 T + p^{5} T^{2} \) |
| 19 | \( 1 + 884 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2268 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1482 T + p^{5} T^{2} \) |
| 31 | \( 1 + 8360 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4714 T + p^{5} T^{2} \) |
| 41 | \( 1 - 9786 T + p^{5} T^{2} \) |
| 43 | \( 1 + 452 p T + p^{5} T^{2} \) |
| 47 | \( 1 - 22200 T + p^{5} T^{2} \) |
| 53 | \( 1 + 26790 T + p^{5} T^{2} \) |
| 59 | \( 1 - 28092 T + p^{5} T^{2} \) |
| 61 | \( 1 + 38866 T + p^{5} T^{2} \) |
| 67 | \( 1 + 23948 T + p^{5} T^{2} \) |
| 71 | \( 1 + 20628 T + p^{5} T^{2} \) |
| 73 | \( 1 - 290 T + p^{5} T^{2} \) |
| 79 | \( 1 - 99544 T + p^{5} T^{2} \) |
| 83 | \( 1 - 19308 T + p^{5} T^{2} \) |
| 89 | \( 1 + 36390 T + p^{5} T^{2} \) |
| 97 | \( 1 + 79078 T + p^{5} T^{2} \) |
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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
−8.975841194989043934455910997280, −8.158907116723062076140859306815, −6.86406213221405836908371825764, −6.22361835429226416018918099370, −5.62943424280289244566848874695, −4.29678735839643011788345927947, −3.49515852852088336164371963628, −2.17221309357490688971466342634, −1.41197995458804473034080927152, 0,
1.41197995458804473034080927152, 2.17221309357490688971466342634, 3.49515852852088336164371963628, 4.29678735839643011788345927947, 5.62943424280289244566848874695, 6.22361835429226416018918099370, 6.86406213221405836908371825764, 8.158907116723062076140859306815, 8.975841194989043934455910997280
