L(s) = 1 | − 2.62·2-s + 3-s + 4.89·4-s − 2.62·6-s − 5.24·7-s − 7.58·8-s + 9-s + 2.45·11-s + 4.89·12-s + 1.47·13-s + 13.7·14-s + 10.1·16-s − 4.14·17-s − 2.62·18-s + 4.25·19-s − 5.24·21-s − 6.44·22-s + 6.50·23-s − 7.58·24-s − 3.88·26-s + 27-s − 25.6·28-s − 8.17·29-s − 5.69·31-s − 11.4·32-s + 2.45·33-s + 10.8·34-s + ⋯ |
L(s) = 1 | − 1.85·2-s + 0.577·3-s + 2.44·4-s − 1.07·6-s − 1.98·7-s − 2.68·8-s + 0.333·9-s + 0.739·11-s + 1.41·12-s + 0.410·13-s + 3.68·14-s + 2.53·16-s − 1.00·17-s − 0.618·18-s + 0.975·19-s − 1.14·21-s − 1.37·22-s + 1.35·23-s − 1.54·24-s − 0.761·26-s + 0.192·27-s − 4.85·28-s − 1.51·29-s − 1.02·31-s − 2.02·32-s + 0.427·33-s + 1.86·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 7 | \( 1 + 5.24T + 7T^{2} \) |
| 11 | \( 1 - 2.45T + 11T^{2} \) |
| 13 | \( 1 - 1.47T + 13T^{2} \) |
| 17 | \( 1 + 4.14T + 17T^{2} \) |
| 19 | \( 1 - 4.25T + 19T^{2} \) |
| 23 | \( 1 - 6.50T + 23T^{2} \) |
| 29 | \( 1 + 8.17T + 29T^{2} \) |
| 31 | \( 1 + 5.69T + 31T^{2} \) |
| 37 | \( 1 - 1.84T + 37T^{2} \) |
| 41 | \( 1 + 9.39T + 41T^{2} \) |
| 43 | \( 1 - 4.34T + 43T^{2} \) |
| 53 | \( 1 - 6.95T + 53T^{2} \) |
| 59 | \( 1 + 7.70T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 - 5.84T + 67T^{2} \) |
| 71 | \( 1 + 2.81T + 71T^{2} \) |
| 73 | \( 1 + 5.10T + 73T^{2} \) |
| 79 | \( 1 - 5.56T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 3.59T + 89T^{2} \) |
| 97 | \( 1 - 3.34T + 97T^{2} \) |
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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
−8.585759014849477990922304496201, −7.39155857823800465381885804236, −7.01644141405851951777582894087, −6.46746464974038463540750242289, −5.59354729635058132466373326668, −3.81216757145875168860377187675, −3.19296087155105010772192928363, −2.33393397010475608769293471420, −1.18794120054339960493656086797, 0,
1.18794120054339960493656086797, 2.33393397010475608769293471420, 3.19296087155105010772192928363, 3.81216757145875168860377187675, 5.59354729635058132466373326668, 6.46746464974038463540750242289, 7.01644141405851951777582894087, 7.39155857823800465381885804236, 8.585759014849477990922304496201