L(s) = 1 | + 0.502·5-s + 7-s + 3.41·11-s + 4.73·13-s − 6.89·17-s + 7.51·19-s − 23-s − 4.74·25-s + 0.637·29-s + 5.23·31-s + 0.502·35-s − 5.84·37-s + 5.06·41-s + 4.37·43-s + 11.6·47-s + 49-s − 2.12·53-s + 1.71·55-s + 12.4·59-s − 11.3·61-s + 2.37·65-s − 1.36·67-s + 11.0·71-s + 14.0·73-s + 3.41·77-s − 16.2·79-s − 14.6·83-s + ⋯ |
L(s) = 1 | + 0.224·5-s + 0.377·7-s + 1.03·11-s + 1.31·13-s − 1.67·17-s + 1.72·19-s − 0.208·23-s − 0.949·25-s + 0.118·29-s + 0.940·31-s + 0.0849·35-s − 0.960·37-s + 0.790·41-s + 0.667·43-s + 1.69·47-s + 0.142·49-s − 0.292·53-s + 0.231·55-s + 1.61·59-s − 1.45·61-s + 0.294·65-s − 0.167·67-s + 1.31·71-s + 1.64·73-s + 0.389·77-s − 1.82·79-s − 1.61·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.582257267\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.582257267\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 0.502T + 5T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 - 4.73T + 13T^{2} \) |
| 17 | \( 1 + 6.89T + 17T^{2} \) |
| 19 | \( 1 - 7.51T + 19T^{2} \) |
| 29 | \( 1 - 0.637T + 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 + 5.84T + 37T^{2} \) |
| 41 | \( 1 - 5.06T + 41T^{2} \) |
| 43 | \( 1 - 4.37T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 2.12T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 1.36T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 4.59T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.203017345548456638582505654313, −7.34240145967444582316841883428, −6.65942821414294969874650008013, −5.99867488330127222876061101264, −5.34022400429200844598937843816, −4.26438661412831363362132292119, −3.85804828535245319911366344252, −2.76885159470983757150324781072, −1.75335054080206791594906327708, −0.911191153798897478834095819448,
0.911191153798897478834095819448, 1.75335054080206791594906327708, 2.76885159470983757150324781072, 3.85804828535245319911366344252, 4.26438661412831363362132292119, 5.34022400429200844598937843816, 5.99867488330127222876061101264, 6.65942821414294969874650008013, 7.34240145967444582316841883428, 8.203017345548456638582505654313