L(s) = 1 | − 2-s − 1.95·3-s + 4-s − 0.613·5-s + 1.95·6-s − 0.465·7-s − 8-s + 0.829·9-s + 0.613·10-s − 4.22·11-s − 1.95·12-s + 0.465·14-s + 1.20·15-s + 16-s + 5.33·17-s − 0.829·18-s − 19-s − 0.613·20-s + 0.911·21-s + 4.22·22-s − 6.57·23-s + 1.95·24-s − 4.62·25-s + 4.24·27-s − 0.465·28-s + 8.11·29-s − 1.20·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.12·3-s + 0.5·4-s − 0.274·5-s + 0.798·6-s − 0.176·7-s − 0.353·8-s + 0.276·9-s + 0.194·10-s − 1.27·11-s − 0.564·12-s + 0.124·14-s + 0.310·15-s + 0.250·16-s + 1.29·17-s − 0.195·18-s − 0.229·19-s − 0.137·20-s + 0.198·21-s + 0.900·22-s − 1.37·23-s + 0.399·24-s − 0.924·25-s + 0.817·27-s − 0.0880·28-s + 1.50·29-s − 0.219·30-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
\(L(1)\) |
\(\approx\) |
\(0.3133638636\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3133638636\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.95T + 3T^{2} \) |
| 5 | \( 1 + 0.613T + 5T^{2} \) |
| 7 | \( 1 + 0.465T + 7T^{2} \) |
| 11 | \( 1 + 4.22T + 11T^{2} \) |
| 17 | \( 1 - 5.33T + 17T^{2} \) |
| 23 | \( 1 + 6.57T + 23T^{2} \) |
| 29 | \( 1 - 8.11T + 29T^{2} \) |
| 31 | \( 1 - 4.04T + 31T^{2} \) |
| 37 | \( 1 + 1.69T + 37T^{2} \) |
| 41 | \( 1 + 4.71T + 41T^{2} \) |
| 43 | \( 1 - 2.48T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 + 3.74T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 4.02T + 61T^{2} \) |
| 67 | \( 1 + 9.47T + 67T^{2} \) |
| 71 | \( 1 - 16.6T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 6.66T + 79T^{2} \) |
| 83 | \( 1 + 6.70T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.057411074075015535905290655565, −7.48617212553629809787206005013, −6.36865085175089726798862544683, −6.15635127463988669672125713368, −5.23493032339159060530081852284, −4.69786012209156414272136599264, −3.47184103898271154365743638056, −2.69973989451488256524276051494, −1.53547444577260145763153842093, −0.34500317277520716273995184808,
0.34500317277520716273995184808, 1.53547444577260145763153842093, 2.69973989451488256524276051494, 3.47184103898271154365743638056, 4.69786012209156414272136599264, 5.23493032339159060530081852284, 6.15635127463988669672125713368, 6.36865085175089726798862544683, 7.48617212553629809787206005013, 8.057411074075015535905290655565