| L(s) = 1 | − 5-s + 4·7-s + 3·11-s − 4·13-s − 5·19-s − 6·23-s + 25-s + 9·29-s − 5·31-s − 4·35-s + 2·37-s + 9·41-s + 10·43-s − 6·47-s + 9·49-s + 12·53-s − 3·55-s + 9·59-s − 10·61-s + 4·65-s − 2·67-s + 3·71-s − 4·73-s + 12·77-s + 4·79-s + 6·83-s + 9·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s + 0.904·11-s − 1.10·13-s − 1.14·19-s − 1.25·23-s + 1/5·25-s + 1.67·29-s − 0.898·31-s − 0.676·35-s + 0.328·37-s + 1.40·41-s + 1.52·43-s − 0.875·47-s + 9/7·49-s + 1.64·53-s − 0.404·55-s + 1.17·59-s − 1.28·61-s + 0.496·65-s − 0.244·67-s + 0.356·71-s − 0.468·73-s + 1.36·77-s + 0.450·79-s + 0.658·83-s + 0.953·89-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
| \(L(1)\) |
\(\approx\) |
\(2.124539789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.124539789\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.85696970039743662579523892425, −7.56443444834111607428372811590, −6.63504589856883711658630800028, −5.91552438875991582051599325133, −4.97343946571293476626972025188, −4.38214856582138122355190677700, −3.90456986137893120399932831529, −2.54397296680307529609827786424, −1.89593164292308179902160059883, −0.76512959571159522590216400215,
0.76512959571159522590216400215, 1.89593164292308179902160059883, 2.54397296680307529609827786424, 3.90456986137893120399932831529, 4.38214856582138122355190677700, 4.97343946571293476626972025188, 5.91552438875991582051599325133, 6.63504589856883711658630800028, 7.56443444834111607428372811590, 7.85696970039743662579523892425
