| L(s) = 1 | + 2-s + 4-s + 1.73·5-s − 3.46·7-s + 8-s + 1.73·10-s + 1.73·13-s − 3.46·14-s + 16-s − 3·17-s − 3.46·19-s + 1.73·20-s + 8.66·23-s − 2.00·25-s + 1.73·26-s − 3.46·28-s − 6·29-s − 8·31-s + 32-s − 3·34-s − 5.99·35-s − 2·37-s − 3.46·38-s + 1.73·40-s − 3·41-s − 10.3·43-s + 8.66·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.774·5-s − 1.30·7-s + 0.353·8-s + 0.547·10-s + 0.480·13-s − 0.925·14-s + 0.250·16-s − 0.727·17-s − 0.794·19-s + 0.387·20-s + 1.80·23-s − 0.400·25-s + 0.339·26-s − 0.654·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.514·34-s − 1.01·35-s − 0.328·37-s − 0.561·38-s + 0.273·40-s − 0.468·41-s − 1.58·43-s + 1.27·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 13 | \( 1 - 1.73T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 - 8.66T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 5.19T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 8.66T + 61T^{2} \) |
| 67 | \( 1 + 5T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 - 3.46T + 73T^{2} \) |
| 79 | \( 1 + 1.73T + 79T^{2} \) |
| 83 | \( 1 + 15T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−7.27582332927882918322844253769, −6.77650225823666460822446013417, −6.25795396031077140465979194254, −5.54036056200593700400154818924, −4.92233529801527245974737138573, −3.80846826121077620582704323379, −3.34647344927780818210173039837, −2.40061568461875204881731068969, −1.59151640259966826293777345481, 0,
1.59151640259966826293777345481, 2.40061568461875204881731068969, 3.34647344927780818210173039837, 3.80846826121077620582704323379, 4.92233529801527245974737138573, 5.54036056200593700400154818924, 6.25795396031077140465979194254, 6.77650225823666460822446013417, 7.27582332927882918322844253769
