L(s) = 1 | + 1.08·3-s + 3.67·5-s − 1.62·7-s − 1.81·9-s + 0.444·11-s + 6.96·13-s + 4.00·15-s − 2.64·17-s − 4.48·19-s − 1.76·21-s + 7.32·23-s + 8.51·25-s − 5.24·27-s − 6.81·29-s − 5.57·31-s + 0.483·33-s − 5.97·35-s + 1.88·37-s + 7.57·39-s + 2.69·41-s + 3.72·43-s − 6.67·45-s + 9.38·47-s − 4.35·49-s − 2.87·51-s + 5.46·53-s + 1.63·55-s + ⋯ |
L(s) = 1 | + 0.628·3-s + 1.64·5-s − 0.614·7-s − 0.605·9-s + 0.133·11-s + 1.93·13-s + 1.03·15-s − 0.641·17-s − 1.02·19-s − 0.385·21-s + 1.52·23-s + 1.70·25-s − 1.00·27-s − 1.26·29-s − 1.00·31-s + 0.0841·33-s − 1.00·35-s + 0.309·37-s + 1.21·39-s + 0.420·41-s + 0.567·43-s − 0.995·45-s + 1.36·47-s − 0.622·49-s − 0.402·51-s + 0.750·53-s + 0.220·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.516189758\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.516189758\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 - 1.08T + 3T^{2} \) |
| 5 | \( 1 - 3.67T + 5T^{2} \) |
| 7 | \( 1 + 1.62T + 7T^{2} \) |
| 11 | \( 1 - 0.444T + 11T^{2} \) |
| 13 | \( 1 - 6.96T + 13T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 19 | \( 1 + 4.48T + 19T^{2} \) |
| 23 | \( 1 - 7.32T + 23T^{2} \) |
| 29 | \( 1 + 6.81T + 29T^{2} \) |
| 31 | \( 1 + 5.57T + 31T^{2} \) |
| 37 | \( 1 - 1.88T + 37T^{2} \) |
| 41 | \( 1 - 2.69T + 41T^{2} \) |
| 43 | \( 1 - 3.72T + 43T^{2} \) |
| 47 | \( 1 - 9.38T + 47T^{2} \) |
| 53 | \( 1 - 5.46T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 6.74T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 + 0.377T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 1.00T + 83T^{2} \) |
| 89 | \( 1 + 6.79T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.020323438332218482353728696939, −6.82537006633777402876116049407, −6.51519116168264394422291691160, −5.65803033844190767185860720509, −5.43762968287175976897401642263, −4.02894950658370007200051300304, −3.47475252380950142603384531508, −2.49486218080901723308996361011, −2.01195142160285321115617642883, −0.911489607024655234161881728016,
0.911489607024655234161881728016, 2.01195142160285321115617642883, 2.49486218080901723308996361011, 3.47475252380950142603384531508, 4.02894950658370007200051300304, 5.43762968287175976897401642263, 5.65803033844190767185860720509, 6.51519116168264394422291691160, 6.82537006633777402876116049407, 8.020323438332218482353728696939