L(s) = 1 | + 0.412·2-s + 2.39·3-s − 1.82·4-s + 5-s + 0.988·6-s + 0.00790·7-s − 1.58·8-s + 2.73·9-s + 0.412·10-s + 5.71·11-s − 4.37·12-s − 4.38·13-s + 0.00326·14-s + 2.39·15-s + 3.00·16-s + 6.42·17-s + 1.12·18-s + 6.30·19-s − 1.82·20-s + 0.0189·21-s + 2.35·22-s − 6.16·23-s − 3.78·24-s + 25-s − 1.80·26-s − 0.644·27-s − 0.0144·28-s + ⋯ |
L(s) = 1 | + 0.291·2-s + 1.38·3-s − 0.914·4-s + 0.447·5-s + 0.403·6-s + 0.00298·7-s − 0.558·8-s + 0.910·9-s + 0.130·10-s + 1.72·11-s − 1.26·12-s − 1.21·13-s + 0.000872·14-s + 0.618·15-s + 0.751·16-s + 1.55·17-s + 0.265·18-s + 1.44·19-s − 0.409·20-s + 0.00413·21-s + 0.502·22-s − 1.28·23-s − 0.772·24-s + 0.200·25-s − 0.354·26-s − 0.123·27-s − 0.00273·28-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
\(L(1)\) |
\(\approx\) |
\(3.133832342\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.133832342\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.412T + 2T^{2} \) |
| 3 | \( 1 - 2.39T + 3T^{2} \) |
| 7 | \( 1 - 0.00790T + 7T^{2} \) |
| 11 | \( 1 - 5.71T + 11T^{2} \) |
| 13 | \( 1 + 4.38T + 13T^{2} \) |
| 17 | \( 1 - 6.42T + 17T^{2} \) |
| 19 | \( 1 - 6.30T + 19T^{2} \) |
| 23 | \( 1 + 6.16T + 23T^{2} \) |
| 29 | \( 1 - 1.39T + 29T^{2} \) |
| 31 | \( 1 + 6.59T + 31T^{2} \) |
| 37 | \( 1 - 5.00T + 37T^{2} \) |
| 41 | \( 1 + 8.98T + 41T^{2} \) |
| 43 | \( 1 - 4.00T + 43T^{2} \) |
| 47 | \( 1 - 2.83T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 8.06T + 59T^{2} \) |
| 61 | \( 1 - 8.61T + 61T^{2} \) |
| 67 | \( 1 - 8.89T + 67T^{2} \) |
| 71 | \( 1 - 4.60T + 71T^{2} \) |
| 73 | \( 1 + 3.97T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 5.02T + 83T^{2} \) |
| 89 | \( 1 - 0.897T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−9.322749891601465279316155248006, −8.451735864149580603260990367435, −7.77759900039164644426811743187, −6.98539704886823486214135806937, −5.77908852702016821140715445399, −5.09554359687383742554179244457, −3.84404873053217332607340766155, −3.55165023267599914711824260403, −2.39945432382167853442686343203, −1.17363201830473263051318850518,
1.17363201830473263051318850518, 2.39945432382167853442686343203, 3.55165023267599914711824260403, 3.84404873053217332607340766155, 5.09554359687383742554179244457, 5.77908852702016821140715445399, 6.98539704886823486214135806937, 7.77759900039164644426811743187, 8.451735864149580603260990367435, 9.322749891601465279316155248006