| L(s) = 1 | − 2-s − 1.57·3-s + 4-s − 1.25·5-s + 1.57·6-s + 0.329·7-s − 8-s − 0.504·9-s + 1.25·10-s − 2.98·11-s − 1.57·12-s − 0.329·14-s + 1.98·15-s + 16-s − 5.34·17-s + 0.504·18-s + 19-s − 1.25·20-s − 0.520·21-s + 2.98·22-s + 2.43·23-s + 1.57·24-s − 3.41·25-s + 5.53·27-s + 0.329·28-s − 7.29·29-s − 1.98·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.912·3-s + 0.5·4-s − 0.562·5-s + 0.644·6-s + 0.124·7-s − 0.353·8-s − 0.168·9-s + 0.397·10-s − 0.899·11-s − 0.456·12-s − 0.0880·14-s + 0.513·15-s + 0.250·16-s − 1.29·17-s + 0.118·18-s + 0.229·19-s − 0.281·20-s − 0.113·21-s + 0.636·22-s + 0.508·23-s + 0.322·24-s − 0.683·25-s + 1.06·27-s + 0.0622·28-s − 1.35·29-s − 0.362·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1613008254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1613008254\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.57T + 3T^{2} \) |
| 5 | \( 1 + 1.25T + 5T^{2} \) |
| 7 | \( 1 - 0.329T + 7T^{2} \) |
| 11 | \( 1 + 2.98T + 11T^{2} \) |
| 17 | \( 1 + 5.34T + 17T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 + 7.29T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 + 7.90T + 37T^{2} \) |
| 41 | \( 1 + 3.76T + 41T^{2} \) |
| 43 | \( 1 + 0.248T + 43T^{2} \) |
| 47 | \( 1 - 2.69T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 0.687T + 61T^{2} \) |
| 67 | \( 1 + 3.71T + 67T^{2} \) |
| 71 | \( 1 + 1.80T + 71T^{2} \) |
| 73 | \( 1 - 4.18T + 73T^{2} \) |
| 79 | \( 1 + 9.63T + 79T^{2} \) |
| 83 | \( 1 + 8.51T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 7.41T + 97T^{2} \) |
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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
−8.107008862072666756846879673562, −7.26568486085878002818924660480, −6.77707325451642606024652161161, −5.90334484757314667089413350143, −5.29308717813998547634256045025, −4.57881153266876389929187113107, −3.55248014945579915373927940745, −2.62924359557619844153358544546, −1.63688634909538246786972336446, −0.23669527309720202602567776507,
0.23669527309720202602567776507, 1.63688634909538246786972336446, 2.62924359557619844153358544546, 3.55248014945579915373927940745, 4.57881153266876389929187113107, 5.29308717813998547634256045025, 5.90334484757314667089413350143, 6.77707325451642606024652161161, 7.26568486085878002818924660480, 8.107008862072666756846879673562
