| L(s) = 1 | + 2·7-s − 6·11-s + 13-s − 17-s + 2·19-s + 6·23-s + 9·29-s − 9·31-s + 12·37-s − 6·41-s + 6·43-s + 12·47-s − 3·49-s − 3·53-s − 7·59-s − 6·61-s − 13·67-s − 10·71-s − 14·73-s − 12·77-s − 2·83-s − 15·89-s + 2·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.80·11-s + 0.277·13-s − 0.242·17-s + 0.458·19-s + 1.25·23-s + 1.67·29-s − 1.61·31-s + 1.97·37-s − 0.937·41-s + 0.914·43-s + 1.75·47-s − 3/7·49-s − 0.412·53-s − 0.911·59-s − 0.768·61-s − 1.58·67-s − 1.18·71-s − 1.63·73-s − 1.36·77-s − 0.219·83-s − 1.58·89-s + 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.904241038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.904241038\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−12.46722146742482, −11.99206295848692, −11.44487453825519, −11.00020352552455, −10.69537475106414, −10.34358227920999, −9.808484517043078, −9.134712291074033, −8.833478306778173, −8.355764786172923, −7.782172604182023, −7.408007324755611, −7.252005586673553, −6.297566034447729, −5.958358219106458, −5.365910513897273, −5.020170726330276, −4.482726070393032, −4.162604444423491, −3.132529228128665, −2.885945857506560, −2.452733658581108, −1.618391763516025, −1.153208110003053, −0.3646259738150699,
0.3646259738150699, 1.153208110003053, 1.618391763516025, 2.452733658581108, 2.885945857506560, 3.132529228128665, 4.162604444423491, 4.482726070393032, 5.020170726330276, 5.365910513897273, 5.958358219106458, 6.297566034447729, 7.252005586673553, 7.408007324755611, 7.782172604182023, 8.355764786172923, 8.833478306778173, 9.134712291074033, 9.808484517043078, 10.34358227920999, 10.69537475106414, 11.00020352552455, 11.44487453825519, 11.99206295848692, 12.46722146742482
