| L(s) = 1 | − 2·4-s − 3·7-s − 4·11-s + 13-s + 4·16-s + 4·17-s − 5·19-s − 4·23-s + 6·28-s + 8·29-s + 5·31-s + 37-s + 8·41-s + 9·43-s + 8·44-s − 12·47-s + 2·49-s − 2·52-s − 4·59-s − 61-s − 8·64-s + 5·67-s − 8·68-s + 4·71-s + 6·73-s + 10·76-s + 12·77-s + ⋯ |
| L(s) = 1 | − 4-s − 1.13·7-s − 1.20·11-s + 0.277·13-s + 16-s + 0.970·17-s − 1.14·19-s − 0.834·23-s + 1.13·28-s + 1.48·29-s + 0.898·31-s + 0.164·37-s + 1.24·41-s + 1.37·43-s + 1.20·44-s − 1.75·47-s + 2/7·49-s − 0.277·52-s − 0.520·59-s − 0.128·61-s − 64-s + 0.610·67-s − 0.970·68-s + 0.474·71-s + 0.702·73-s + 1.14·76-s + 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.82739137911933209347770335441, −6.52975259644028179159347428552, −6.14489688041184178763690113382, −5.34307551255757580880859691045, −4.62611324182328316046259443344, −3.90780258682550720461373833891, −3.12930490596597876042466594411, −2.42840439950484648016631405052, −0.945396842443061788002559149516, 0,
0.945396842443061788002559149516, 2.42840439950484648016631405052, 3.12930490596597876042466594411, 3.90780258682550720461373833891, 4.62611324182328316046259443344, 5.34307551255757580880859691045, 6.14489688041184178763690113382, 6.52975259644028179159347428552, 7.82739137911933209347770335441
