| L(s) = 1 | + 2.82·5-s − 3·9-s + 11-s + 1.41·13-s − 2.82·17-s + 4.24·19-s − 2·23-s + 3.00·25-s + 1.41·31-s + 2·37-s + 8.48·41-s + 2·43-s − 8.48·45-s + 7.07·47-s + 6·53-s + 2.82·55-s + 2.82·59-s − 1.41·61-s + 4.00·65-s − 4·67-s + 8·71-s − 2.82·73-s + 8·79-s + 9·81-s + 1.41·83-s − 8.00·85-s − 7.07·89-s + ⋯ |
| L(s) = 1 | + 1.26·5-s − 9-s + 0.301·11-s + 0.392·13-s − 0.685·17-s + 0.973·19-s − 0.417·23-s + 0.600·25-s + 0.254·31-s + 0.328·37-s + 1.32·41-s + 0.304·43-s − 1.26·45-s + 1.03·47-s + 0.824·53-s + 0.381·55-s + 0.368·59-s − 0.181·61-s + 0.496·65-s − 0.488·67-s + 0.949·71-s − 0.331·73-s + 0.900·79-s + 81-s + 0.155·83-s − 0.867·85-s − 0.749·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.391219973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.391219973\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 7.07T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 2.82T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 1.41T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.532904096310087730839510218477, −7.67189681264346163049386894282, −6.79130393741163844886731232737, −5.92338872962367287017193450967, −5.74146790258974067214522469731, −4.75592611428072030383994695302, −3.75138966044196732849838987303, −2.73708455987763734577600487758, −2.07043812471889324616313471760, −0.883996991495741715479569187060,
0.883996991495741715479569187060, 2.07043812471889324616313471760, 2.73708455987763734577600487758, 3.75138966044196732849838987303, 4.75592611428072030383994695302, 5.74146790258974067214522469731, 5.92338872962367287017193450967, 6.79130393741163844886731232737, 7.67189681264346163049386894282, 8.532904096310087730839510218477
