| L(s) = 1 | − 2-s − 0.416·3-s + 4-s + 0.917·5-s + 0.416·6-s − 0.743·7-s − 8-s − 2.82·9-s − 0.917·10-s − 5.10·11-s − 0.416·12-s + 4.45·13-s + 0.743·14-s − 0.381·15-s + 16-s − 2.36·17-s + 2.82·18-s − 0.965·19-s + 0.917·20-s + 0.309·21-s + 5.10·22-s − 4.94·23-s + 0.416·24-s − 4.15·25-s − 4.45·26-s + 2.42·27-s − 0.743·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.240·3-s + 0.5·4-s + 0.410·5-s + 0.169·6-s − 0.280·7-s − 0.353·8-s − 0.942·9-s − 0.290·10-s − 1.53·11-s − 0.120·12-s + 1.23·13-s + 0.198·14-s − 0.0985·15-s + 0.250·16-s − 0.572·17-s + 0.666·18-s − 0.221·19-s + 0.205·20-s + 0.0674·21-s + 1.08·22-s − 1.03·23-s + 0.0849·24-s − 0.831·25-s − 0.872·26-s + 0.466·27-s − 0.140·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6717552064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6717552064\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 0.416T + 3T^{2} \) |
| 5 | \( 1 - 0.917T + 5T^{2} \) |
| 7 | \( 1 + 0.743T + 7T^{2} \) |
| 11 | \( 1 + 5.10T + 11T^{2} \) |
| 13 | \( 1 - 4.45T + 13T^{2} \) |
| 17 | \( 1 + 2.36T + 17T^{2} \) |
| 19 | \( 1 + 0.965T + 19T^{2} \) |
| 23 | \( 1 + 4.94T + 23T^{2} \) |
| 29 | \( 1 - 0.0789T + 29T^{2} \) |
| 37 | \( 1 + 0.681T + 37T^{2} \) |
| 41 | \( 1 + 2.29T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 6.22T + 47T^{2} \) |
| 53 | \( 1 + 7.21T + 53T^{2} \) |
| 59 | \( 1 - 1.52T + 59T^{2} \) |
| 61 | \( 1 - 9.07T + 61T^{2} \) |
| 67 | \( 1 + 5.30T + 67T^{2} \) |
| 71 | \( 1 + 9.12T + 71T^{2} \) |
| 73 | \( 1 - 8.26T + 73T^{2} \) |
| 79 | \( 1 + 8.79T + 79T^{2} \) |
| 83 | \( 1 + 3.04T + 83T^{2} \) |
| 89 | \( 1 + 4.63T + 89T^{2} \) |
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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
−8.180896648179599496449788802541, −7.56023967212317732743823978666, −6.60074438966340030643267952121, −5.80398231919067315827167507237, −5.68864558776129330126982637682, −4.47928259326960258239162866761, −3.43570616599156112631905077582, −2.59822748646229331929477814978, −1.87098094452409934983642059647, −0.46024241388729130406893832579,
0.46024241388729130406893832579, 1.87098094452409934983642059647, 2.59822748646229331929477814978, 3.43570616599156112631905077582, 4.47928259326960258239162866761, 5.68864558776129330126982637682, 5.80398231919067315827167507237, 6.60074438966340030643267952121, 7.56023967212317732743823978666, 8.180896648179599496449788802541
