| L(s) = 1 | − 5·7-s + 6·11-s − 3·13-s + 2·17-s + 19-s + 2·23-s − 6·29-s + 3·31-s − 6·37-s − 4·41-s + 11·43-s + 10·47-s + 18·49-s + 8·53-s + 6·59-s + 3·61-s − 67-s + 12·71-s + 10·73-s − 30·77-s − 8·79-s + 6·83-s + 16·89-s + 15·91-s − 7·97-s + 8·101-s + 4·103-s + ⋯ |
| L(s) = 1 | − 1.88·7-s + 1.80·11-s − 0.832·13-s + 0.485·17-s + 0.229·19-s + 0.417·23-s − 1.11·29-s + 0.538·31-s − 0.986·37-s − 0.624·41-s + 1.67·43-s + 1.45·47-s + 18/7·49-s + 1.09·53-s + 0.781·59-s + 0.384·61-s − 0.122·67-s + 1.42·71-s + 1.17·73-s − 3.41·77-s − 0.900·79-s + 0.658·83-s + 1.69·89-s + 1.57·91-s − 0.710·97-s + 0.796·101-s + 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.392251311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.392251311\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−9.321220473769028887363483622390, −8.824682847220706974237362604525, −7.41988521419286836006930734564, −6.87286738401673998834130048890, −6.20813236608777087639684950878, −5.36594699468031390111821827200, −3.98997962398680768258184211607, −3.50153883662204758784796757836, −2.39119182688794025983111556916, −0.801007844265166692305002560526,
0.801007844265166692305002560526, 2.39119182688794025983111556916, 3.50153883662204758784796757836, 3.98997962398680768258184211607, 5.36594699468031390111821827200, 6.20813236608777087639684950878, 6.87286738401673998834130048890, 7.41988521419286836006930734564, 8.824682847220706974237362604525, 9.321220473769028887363483622390
