| L(s) = 1 | + 5-s + 2·7-s − 2·11-s + 4·13-s + 2·17-s + 4·19-s − 8·23-s + 25-s + 10·29-s + 4·31-s + 2·35-s − 8·43-s − 8·47-s − 3·49-s − 6·53-s − 2·55-s + 14·59-s − 14·61-s + 4·65-s − 4·67-s − 12·71-s + 6·73-s − 4·77-s − 12·79-s − 4·83-s + 2·85-s + 12·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 0.603·11-s + 1.10·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s + 1.85·29-s + 0.718·31-s + 0.338·35-s − 1.21·43-s − 1.16·47-s − 3/7·49-s − 0.824·53-s − 0.269·55-s + 1.82·59-s − 1.79·61-s + 0.496·65-s − 0.488·67-s − 1.42·71-s + 0.702·73-s − 0.455·77-s − 1.35·79-s − 0.439·83-s + 0.216·85-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.568979760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.568979760\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−11.48759471734628960075235640364, −10.42977374464218618259946458906, −9.789609061778077574844038624674, −8.430531363871330384753929024064, −7.942341734598117376456164094334, −6.53876572552795316440603698045, −5.59184082567252664246215870943, −4.54321125337695258384863067085, −3.08000224242427569012154075457, −1.50557401701584832617156823174,
1.50557401701584832617156823174, 3.08000224242427569012154075457, 4.54321125337695258384863067085, 5.59184082567252664246215870943, 6.53876572552795316440603698045, 7.942341734598117376456164094334, 8.430531363871330384753929024064, 9.789609061778077574844038624674, 10.42977374464218618259946458906, 11.48759471734628960075235640364
