| L(s) = 1 | − 1.87·2-s − 0.704·3-s + 1.50·4-s − 5-s + 1.31·6-s + 0.932·8-s − 2.50·9-s + 1.87·10-s + 4.51·11-s − 1.05·12-s + 6.39·13-s + 0.704·15-s − 4.74·16-s + 1.66·17-s + 4.68·18-s + 19-s − 1.50·20-s − 8.44·22-s + 4.99·23-s − 0.657·24-s + 25-s − 11.9·26-s + 3.87·27-s + 3.75·29-s − 1.31·30-s + 8.54·31-s + 7.02·32-s + ⋯ |
| L(s) = 1 | − 1.32·2-s − 0.406·3-s + 0.750·4-s − 0.447·5-s + 0.538·6-s + 0.329·8-s − 0.834·9-s + 0.591·10-s + 1.36·11-s − 0.305·12-s + 1.77·13-s + 0.181·15-s − 1.18·16-s + 0.402·17-s + 1.10·18-s + 0.229·19-s − 0.335·20-s − 1.80·22-s + 1.04·23-s − 0.134·24-s + 0.200·25-s − 2.34·26-s + 0.746·27-s + 0.697·29-s − 0.240·30-s + 1.53·31-s + 1.24·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4655 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4655 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9428552410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9428552410\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.87T + 2T^{2} \) |
| 3 | \( 1 + 0.704T + 3T^{2} \) |
| 11 | \( 1 - 4.51T + 11T^{2} \) |
| 13 | \( 1 - 6.39T + 13T^{2} \) |
| 17 | \( 1 - 1.66T + 17T^{2} \) |
| 23 | \( 1 - 4.99T + 23T^{2} \) |
| 29 | \( 1 - 3.75T + 29T^{2} \) |
| 31 | \( 1 - 8.54T + 31T^{2} \) |
| 37 | \( 1 - 6.01T + 37T^{2} \) |
| 41 | \( 1 - 7.43T + 41T^{2} \) |
| 43 | \( 1 + 8.44T + 43T^{2} \) |
| 47 | \( 1 + 4.68T + 47T^{2} \) |
| 53 | \( 1 - 7.70T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 - 8.22T + 67T^{2} \) |
| 71 | \( 1 - 9.81T + 71T^{2} \) |
| 73 | \( 1 + 5.55T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 + 3.31T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 0.178T + 97T^{2} \) |
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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
−8.326354898135839225613054229966, −7.998700198133921500351735142331, −6.71347930461919421469648955218, −6.55183124889918177650920187535, −5.55656416679619465397168397357, −4.52634568390304830597618875927, −3.73172435503476141224232275427, −2.77491768263216037614613788892, −1.26329159326215093760081653010, −0.825478197685397915856751775695,
0.825478197685397915856751775695, 1.26329159326215093760081653010, 2.77491768263216037614613788892, 3.73172435503476141224232275427, 4.52634568390304830597618875927, 5.55656416679619465397168397357, 6.55183124889918177650920187535, 6.71347930461919421469648955218, 7.998700198133921500351735142331, 8.326354898135839225613054229966
