| L(s) = 1 | − 129.·3-s − 1.99e3·5-s − 7.55e3·7-s − 3.01e3·9-s − 5.33e4·11-s − 2.85e4·13-s + 2.57e5·15-s − 6.80e4·17-s − 7.55e5·19-s + 9.75e5·21-s + 1.43e5·23-s + 2.01e6·25-s + 2.93e6·27-s − 3.05e6·29-s + 4.57e6·31-s + 6.89e6·33-s + 1.50e7·35-s − 1.06e7·37-s + 3.68e6·39-s + 2.61e7·41-s − 4.45e6·43-s + 6.00e6·45-s − 3.57e7·47-s + 1.67e7·49-s + 8.78e6·51-s + 2.60e7·53-s + 1.06e8·55-s + ⋯ |
| L(s) = 1 | − 0.920·3-s − 1.42·5-s − 1.18·7-s − 0.153·9-s − 1.09·11-s − 0.277·13-s + 1.31·15-s − 0.197·17-s − 1.32·19-s + 1.09·21-s + 0.106·23-s + 1.03·25-s + 1.06·27-s − 0.801·29-s + 0.890·31-s + 1.01·33-s + 1.69·35-s − 0.935·37-s + 0.255·39-s + 1.44·41-s − 0.198·43-s + 0.218·45-s − 1.06·47-s + 0.414·49-s + 0.181·51-s + 0.453·53-s + 1.56·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.07122773176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07122773176\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 3 | \( 1 + 129.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.99e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 7.55e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.33e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 6.80e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.55e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.43e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.05e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.57e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.06e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.61e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.45e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.57e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.60e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.91e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 6.51e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.10e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.72e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.54e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.50e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.80e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.95e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.03e9T + 7.60e17T^{2} \) |
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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
−13.10435568388759944158980700363, −12.27989472793451858579840483810, −11.25009371758764514707804140361, −10.28124620064336657844186976550, −8.557811670559285306962274922618, −7.24565508534606980568359928027, −5.98341810902870586456085878662, −4.46901827149790984256529610453, −2.97971168813986961905058956854, −0.16707591072205895864363588926,
0.16707591072205895864363588926, 2.97971168813986961905058956854, 4.46901827149790984256529610453, 5.98341810902870586456085878662, 7.24565508534606980568359928027, 8.557811670559285306962274922618, 10.28124620064336657844186976550, 11.25009371758764514707804140361, 12.27989472793451858579840483810, 13.10435568388759944158980700363
