| L(s) = 1 | + 0.871·2-s − 1.24·4-s + 4.06·5-s + 7-s − 2.82·8-s + 3.54·10-s + 6.09·13-s + 0.871·14-s + 0.0189·16-s + 1.70·17-s − 5.45·19-s − 5.04·20-s + 6.39·23-s + 11.5·25-s + 5.30·26-s − 1.24·28-s + 4.74·29-s − 4.36·31-s + 5.66·32-s + 1.48·34-s + 4.06·35-s + 2.43·37-s − 4.75·38-s − 11.4·40-s − 3.21·41-s − 0.127·43-s + 5.57·46-s + ⋯ |
| L(s) = 1 | + 0.616·2-s − 0.620·4-s + 1.81·5-s + 0.377·7-s − 0.998·8-s + 1.12·10-s + 1.68·13-s + 0.232·14-s + 0.00474·16-s + 0.412·17-s − 1.25·19-s − 1.12·20-s + 1.33·23-s + 2.30·25-s + 1.04·26-s − 0.234·28-s + 0.881·29-s − 0.783·31-s + 1.00·32-s + 0.254·34-s + 0.687·35-s + 0.401·37-s − 0.771·38-s − 1.81·40-s − 0.501·41-s − 0.0193·43-s + 0.821·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.959720370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.959720370\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.871T + 2T^{2} \) |
| 5 | \( 1 - 4.06T + 5T^{2} \) |
| 13 | \( 1 - 6.09T + 13T^{2} \) |
| 17 | \( 1 - 1.70T + 17T^{2} \) |
| 19 | \( 1 + 5.45T + 19T^{2} \) |
| 23 | \( 1 - 6.39T + 23T^{2} \) |
| 29 | \( 1 - 4.74T + 29T^{2} \) |
| 31 | \( 1 + 4.36T + 31T^{2} \) |
| 37 | \( 1 - 2.43T + 37T^{2} \) |
| 41 | \( 1 + 3.21T + 41T^{2} \) |
| 43 | \( 1 + 0.127T + 43T^{2} \) |
| 47 | \( 1 + 8.46T + 47T^{2} \) |
| 53 | \( 1 - 4.71T + 53T^{2} \) |
| 59 | \( 1 + 4.63T + 59T^{2} \) |
| 61 | \( 1 + 5.31T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 + 2.21T + 71T^{2} \) |
| 73 | \( 1 - 5.57T + 73T^{2} \) |
| 79 | \( 1 + 6.27T + 79T^{2} \) |
| 83 | \( 1 + 0.127T + 83T^{2} \) |
| 89 | \( 1 - 8.12T + 89T^{2} \) |
| 97 | \( 1 + 2.59T + 97T^{2} \) |
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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
−8.131046586107142216052303122081, −6.73111372385331555279094351468, −6.35565770996684121979008903282, −5.67865316618704059274674890555, −5.17223184701496458744104791567, −4.46924582314452759382966850914, −3.54874866485497132819468109527, −2.78516884139379287868070721395, −1.77953597960529035707312861298, −0.985139180003050969382983945775,
0.985139180003050969382983945775, 1.77953597960529035707312861298, 2.78516884139379287868070721395, 3.54874866485497132819468109527, 4.46924582314452759382966850914, 5.17223184701496458744104791567, 5.67865316618704059274674890555, 6.35565770996684121979008903282, 6.73111372385331555279094351468, 8.131046586107142216052303122081
