| L(s) = 1 | − 2·7-s + 4·11-s + 13-s + 6·19-s − 6·23-s − 8·29-s + 10·37-s + 10·41-s − 12·43-s − 12·47-s − 3·49-s + 6·53-s + 12·59-s + 10·61-s − 4·67-s + 8·71-s + 4·73-s − 8·77-s − 8·79-s + 4·83-s − 10·89-s − 2·91-s + 4·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1.20·11-s + 0.277·13-s + 1.37·19-s − 1.25·23-s − 1.48·29-s + 1.64·37-s + 1.56·41-s − 1.82·43-s − 1.75·47-s − 3/7·49-s + 0.824·53-s + 1.56·59-s + 1.28·61-s − 0.488·67-s + 0.949·71-s + 0.468·73-s − 0.911·77-s − 0.900·79-s + 0.439·83-s − 1.05·89-s − 0.209·91-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.313651699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.313651699\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.07026819856413, −12.86094960669829, −12.04351732549677, −11.62191892452370, −11.46758162919254, −10.88997293699893, −9.991064402894641, −9.827229306934296, −9.430823823300224, −9.022611686305673, −8.133816040607363, −8.077367436258353, −7.198191483395539, −6.889285403467083, −6.307251470063484, −5.865282329970648, −5.433770144176344, −4.708323383994035, −4.027943174017605, −3.660780873145270, −3.235492133282149, −2.456234388182974, −1.809217484084519, −1.152141831907075, −0.4674959926271464,
0.4674959926271464, 1.152141831907075, 1.809217484084519, 2.456234388182974, 3.235492133282149, 3.660780873145270, 4.027943174017605, 4.708323383994035, 5.433770144176344, 5.865282329970648, 6.307251470063484, 6.889285403467083, 7.198191483395539, 8.077367436258353, 8.133816040607363, 9.022611686305673, 9.430823823300224, 9.827229306934296, 9.991064402894641, 10.88997293699893, 11.46758162919254, 11.62191892452370, 12.04351732549677, 12.86094960669829, 13.07026819856413
