L(s) = 1 | − 2.02·2-s + 1.24·3-s + 2.10·4-s + 5-s − 2.53·6-s + 2.30·7-s − 0.207·8-s − 1.43·9-s − 2.02·10-s − 2.59·11-s + 2.62·12-s − 4.76·13-s − 4.67·14-s + 1.24·15-s − 3.78·16-s − 0.735·17-s + 2.91·18-s + 6.51·19-s + 2.10·20-s + 2.88·21-s + 5.25·22-s + 2.76·23-s − 0.259·24-s + 25-s + 9.65·26-s − 5.54·27-s + 4.85·28-s + ⋯ |
L(s) = 1 | − 1.43·2-s + 0.721·3-s + 1.05·4-s + 0.447·5-s − 1.03·6-s + 0.872·7-s − 0.0735·8-s − 0.479·9-s − 0.640·10-s − 0.782·11-s + 0.758·12-s − 1.32·13-s − 1.24·14-s + 0.322·15-s − 0.946·16-s − 0.178·17-s + 0.687·18-s + 1.49·19-s + 0.470·20-s + 0.629·21-s + 1.12·22-s + 0.577·23-s − 0.0530·24-s + 0.200·25-s + 1.89·26-s − 1.06·27-s + 0.917·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 + 2.02T + 2T^{2} \) |
| 3 | \( 1 - 1.24T + 3T^{2} \) |
| 7 | \( 1 - 2.30T + 7T^{2} \) |
| 11 | \( 1 + 2.59T + 11T^{2} \) |
| 13 | \( 1 + 4.76T + 13T^{2} \) |
| 17 | \( 1 + 0.735T + 17T^{2} \) |
| 19 | \( 1 - 6.51T + 19T^{2} \) |
| 23 | \( 1 - 2.76T + 23T^{2} \) |
| 29 | \( 1 + 9.32T + 29T^{2} \) |
| 31 | \( 1 + 7.30T + 31T^{2} \) |
| 37 | \( 1 + 6.61T + 37T^{2} \) |
| 41 | \( 1 + 2.66T + 41T^{2} \) |
| 43 | \( 1 - 0.405T + 43T^{2} \) |
| 47 | \( 1 + 1.36T + 47T^{2} \) |
| 53 | \( 1 + 9.09T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 1.02T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 + 3.86T + 73T^{2} \) |
| 79 | \( 1 - 17.3T + 79T^{2} \) |
| 83 | \( 1 + 6.59T + 83T^{2} \) |
| 89 | \( 1 + 8.10T + 89T^{2} \) |
| 97 | \( 1 - 9.78T + 97T^{2} \) |
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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
−8.937090334485476683525721451516, −7.82416110237701430953012421110, −7.77921364676443780435082909033, −6.90191281003860052129532165082, −5.39759722093833068610593639589, −4.98780213645376271648357711491, −3.39212698496578333525687625914, −2.34473199323537023563863843907, −1.65024027822502708009171622299, 0,
1.65024027822502708009171622299, 2.34473199323537023563863843907, 3.39212698496578333525687625914, 4.98780213645376271648357711491, 5.39759722093833068610593639589, 6.90191281003860052129532165082, 7.77921364676443780435082909033, 7.82416110237701430953012421110, 8.937090334485476683525721451516