| L(s) = 1 | − 5-s − 2·7-s + 3·11-s − 4·13-s − 6·17-s + 19-s + 6·23-s + 25-s − 9·29-s + 31-s + 2·35-s + 8·37-s + 3·41-s + 4·43-s − 12·47-s − 3·49-s + 6·53-s − 3·55-s − 3·59-s − 10·61-s + 4·65-s − 14·67-s + 3·71-s + 2·73-s − 6·77-s + 16·79-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 0.904·11-s − 1.10·13-s − 1.45·17-s + 0.229·19-s + 1.25·23-s + 1/5·25-s − 1.67·29-s + 0.179·31-s + 0.338·35-s + 1.31·37-s + 0.468·41-s + 0.609·43-s − 1.75·47-s − 3/7·49-s + 0.824·53-s − 0.404·55-s − 0.390·59-s − 1.28·61-s + 0.496·65-s − 1.71·67-s + 0.356·71-s + 0.234·73-s − 0.683·77-s + 1.80·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.159309046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.159309046\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.83093424050918366118058664281, −7.31634732806162182549924766859, −6.60974836809529081483534443352, −6.10452425215086903293524277451, −4.98902518526660590212304514825, −4.43515755464308589855031798954, −3.58793942297082774623358962824, −2.83526549962978887255799847367, −1.88842341154198622491535596635, −0.53973524066983613790810886826,
0.53973524066983613790810886826, 1.88842341154198622491535596635, 2.83526549962978887255799847367, 3.58793942297082774623358962824, 4.43515755464308589855031798954, 4.98902518526660590212304514825, 6.10452425215086903293524277451, 6.60974836809529081483534443352, 7.31634732806162182549924766859, 7.83093424050918366118058664281
