L(s) = 1 | + 3·5-s + 7-s − 6·11-s − 13-s + 8·17-s − 19-s − 23-s + 4·25-s + 5·29-s + 3·31-s + 3·35-s + 12·37-s − 10·41-s − 11·43-s + 3·47-s + 49-s + 53-s − 18·55-s + 8·59-s + 2·61-s − 3·65-s + 2·67-s − 8·71-s + 11·73-s − 6·77-s + 13·79-s + 7·83-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 0.377·7-s − 1.80·11-s − 0.277·13-s + 1.94·17-s − 0.229·19-s − 0.208·23-s + 4/5·25-s + 0.928·29-s + 0.538·31-s + 0.507·35-s + 1.97·37-s − 1.56·41-s − 1.67·43-s + 0.437·47-s + 1/7·49-s + 0.137·53-s − 2.42·55-s + 1.04·59-s + 0.256·61-s − 0.372·65-s + 0.244·67-s − 0.949·71-s + 1.28·73-s − 0.683·77-s + 1.46·79-s + 0.768·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.632398208\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.632398208\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 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
−8.076338535830487059327373832421, −7.40932557585347156543741399302, −6.46212285818300468296740723283, −5.75996429551651431345261301215, −5.22681463243730871841155306889, −4.73229481346635177444876521333, −3.37811428815993667385712448598, −2.63047241791502753917662926061, −1.94786199622203952249066764095, −0.844940816180295573424825489467,
0.844940816180295573424825489467, 1.94786199622203952249066764095, 2.63047241791502753917662926061, 3.37811428815993667385712448598, 4.73229481346635177444876521333, 5.22681463243730871841155306889, 5.75996429551651431345261301215, 6.46212285818300468296740723283, 7.40932557585347156543741399302, 8.076338535830487059327373832421