L(s) = 1 | − 1.04·2-s − 3-s − 0.900·4-s − 0.0290·5-s + 1.04·6-s + 2.65·7-s + 3.04·8-s + 9-s + 0.0304·10-s − 4.86·11-s + 0.900·12-s + 5.69·13-s − 2.78·14-s + 0.0290·15-s − 1.38·16-s − 17-s − 1.04·18-s − 4.01·19-s + 0.0261·20-s − 2.65·21-s + 5.09·22-s + 6.54·23-s − 3.04·24-s − 4.99·25-s − 5.96·26-s − 27-s − 2.38·28-s + ⋯ |
L(s) = 1 | − 0.741·2-s − 0.577·3-s − 0.450·4-s − 0.0130·5-s + 0.428·6-s + 1.00·7-s + 1.07·8-s + 0.333·9-s + 0.00964·10-s − 1.46·11-s + 0.260·12-s + 1.57·13-s − 0.743·14-s + 0.00750·15-s − 0.346·16-s − 0.242·17-s − 0.247·18-s − 0.920·19-s + 0.00585·20-s − 0.578·21-s + 1.08·22-s + 1.36·23-s − 0.620·24-s − 0.999·25-s − 1.17·26-s − 0.192·27-s − 0.451·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 + 1.04T + 2T^{2} \) |
| 5 | \( 1 + 0.0290T + 5T^{2} \) |
| 7 | \( 1 - 2.65T + 7T^{2} \) |
| 11 | \( 1 + 4.86T + 11T^{2} \) |
| 13 | \( 1 - 5.69T + 13T^{2} \) |
| 19 | \( 1 + 4.01T + 19T^{2} \) |
| 23 | \( 1 - 6.54T + 23T^{2} \) |
| 29 | \( 1 - 2.20T + 29T^{2} \) |
| 31 | \( 1 + 4.01T + 31T^{2} \) |
| 37 | \( 1 + 9.08T + 37T^{2} \) |
| 41 | \( 1 - 3.70T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 1.67T + 53T^{2} \) |
| 59 | \( 1 + 9.85T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 5.10T + 67T^{2} \) |
| 71 | \( 1 - 6.23T + 71T^{2} \) |
| 73 | \( 1 + 2.43T + 73T^{2} \) |
| 83 | \( 1 - 9.03T + 83T^{2} \) |
| 89 | \( 1 + 1.57T + 89T^{2} \) |
| 97 | \( 1 + 4.85T + 97T^{2} \) |
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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
−8.274690609237215925834999392562, −7.55503094211631129584096677948, −6.75582141093585248454394303670, −5.68878604074317086031074476141, −5.12692673794793284262176311966, −4.43840909054883975490609827926, −3.53182183778806306382697657828, −2.08397450045565652242909638346, −1.20330767270009280318271605417, 0,
1.20330767270009280318271605417, 2.08397450045565652242909638346, 3.53182183778806306382697657828, 4.43840909054883975490609827926, 5.12692673794793284262176311966, 5.68878604074317086031074476141, 6.75582141093585248454394303670, 7.55503094211631129584096677948, 8.274690609237215925834999392562