| L(s) = 1 | − 4·7-s − 13-s + 17-s + 4·19-s − 8·23-s + 10·29-s − 4·31-s − 10·37-s + 10·41-s + 4·43-s + 8·47-s + 9·49-s − 10·53-s − 4·59-s − 2·61-s + 12·67-s + 12·71-s + 2·73-s + 8·79-s + 4·83-s + 6·89-s + 4·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.277·13-s + 0.242·17-s + 0.917·19-s − 1.66·23-s + 1.85·29-s − 0.718·31-s − 1.64·37-s + 1.56·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s − 1.37·53-s − 0.520·59-s − 0.256·61-s + 1.46·67-s + 1.42·71-s + 0.234·73-s + 0.900·79-s + 0.439·83-s + 0.635·89-s + 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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
| \(L(1)\) |
\(\approx\) |
\(2.204337079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.204337079\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.42699732698133, −12.11412765960895, −11.73339818813374, −10.95920278299762, −10.55010191291941, −10.18003923956537, −9.701583400933812, −9.293387337040271, −9.073456831516002, −8.203686545975407, −7.957318528819451, −7.350465990377150, −6.902930136171408, −6.403023473200424, −6.059108263279009, −5.536644381994171, −5.049399278025104, −4.356966061259283, −3.893623437756177, −3.290185773618956, −3.049905789838435, −2.289247757084467, −1.860789391768908, −0.8037978751951777, −0.5010795116272815,
0.5010795116272815, 0.8037978751951777, 1.860789391768908, 2.289247757084467, 3.049905789838435, 3.290185773618956, 3.893623437756177, 4.356966061259283, 5.049399278025104, 5.536644381994171, 6.059108263279009, 6.403023473200424, 6.902930136171408, 7.350465990377150, 7.957318528819451, 8.203686545975407, 9.073456831516002, 9.293387337040271, 9.701583400933812, 10.18003923956537, 10.55010191291941, 10.95920278299762, 11.73339818813374, 12.11412765960895, 12.42699732698133
