| L(s) = 1 | − 2.69·2-s − 0.494·3-s + 5.28·4-s + 5-s + 1.33·6-s + 3.45·7-s − 8.88·8-s − 2.75·9-s − 2.69·10-s + 2.05·11-s − 2.61·12-s − 4.61·13-s − 9.33·14-s − 0.494·15-s + 13.3·16-s + 0.0376·17-s + 7.43·18-s − 6.34·19-s + 5.28·20-s − 1.71·21-s − 5.54·22-s − 5.72·23-s + 4.39·24-s + 25-s + 12.4·26-s + 2.84·27-s + 18.2·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s − 0.285·3-s + 2.64·4-s + 0.447·5-s + 0.545·6-s + 1.30·7-s − 3.13·8-s − 0.918·9-s − 0.853·10-s + 0.619·11-s − 0.755·12-s − 1.27·13-s − 2.49·14-s − 0.127·15-s + 3.34·16-s + 0.00913·17-s + 1.75·18-s − 1.45·19-s + 1.18·20-s − 0.373·21-s − 1.18·22-s − 1.19·23-s + 0.896·24-s + 0.200·25-s + 2.44·26-s + 0.547·27-s + 3.45·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6026397431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6026397431\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 1607 | \( 1 + T \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 3 | \( 1 + 0.494T + 3T^{2} \) |
| 7 | \( 1 - 3.45T + 7T^{2} \) |
| 11 | \( 1 - 2.05T + 11T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 - 0.0376T + 17T^{2} \) |
| 19 | \( 1 + 6.34T + 19T^{2} \) |
| 23 | \( 1 + 5.72T + 23T^{2} \) |
| 29 | \( 1 - 7.38T + 29T^{2} \) |
| 31 | \( 1 - 0.978T + 31T^{2} \) |
| 37 | \( 1 + 2.58T + 37T^{2} \) |
| 41 | \( 1 + 1.56T + 41T^{2} \) |
| 43 | \( 1 - 2.20T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 1.30T + 59T^{2} \) |
| 61 | \( 1 + 8.58T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 0.0361T + 71T^{2} \) |
| 73 | \( 1 + 6.51T + 73T^{2} \) |
| 79 | \( 1 - 8.52T + 79T^{2} \) |
| 83 | \( 1 - 6.10T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 6.54T + 97T^{2} \) |
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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
−7.956000103825223624860870462898, −7.47076580440977088371692267677, −6.47214540894659682997723842644, −6.21457852502014666640685702763, −5.20975962498479944212032935060, −4.42889212809908715840031557865, −2.98550635098970783098047000403, −2.16146219778777377519014174547, −1.69208951581683417634821273634, −0.51032518548828927404525772639,
0.51032518548828927404525772639, 1.69208951581683417634821273634, 2.16146219778777377519014174547, 2.98550635098970783098047000403, 4.42889212809908715840031557865, 5.20975962498479944212032935060, 6.21457852502014666640685702763, 6.47214540894659682997723842644, 7.47076580440977088371692267677, 7.956000103825223624860870462898
