L(s) = 1 | + 2.22·2-s − 1.47·3-s + 2.96·4-s + 5-s − 3.29·6-s − 4.42·7-s + 2.15·8-s − 0.812·9-s + 2.22·10-s − 11-s − 4.38·12-s − 3.92·13-s − 9.86·14-s − 1.47·15-s − 1.13·16-s − 1.61·17-s − 1.81·18-s + 19-s + 2.96·20-s + 6.54·21-s − 2.22·22-s + 0.113·23-s − 3.18·24-s + 25-s − 8.75·26-s + 5.63·27-s − 13.1·28-s + ⋯ |
L(s) = 1 | + 1.57·2-s − 0.853·3-s + 1.48·4-s + 0.447·5-s − 1.34·6-s − 1.67·7-s + 0.762·8-s − 0.270·9-s + 0.704·10-s − 0.301·11-s − 1.26·12-s − 1.08·13-s − 2.63·14-s − 0.381·15-s − 0.282·16-s − 0.391·17-s − 0.427·18-s + 0.229·19-s + 0.663·20-s + 1.42·21-s − 0.475·22-s + 0.0236·23-s − 0.650·24-s + 0.200·25-s − 1.71·26-s + 1.08·27-s − 2.48·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 - 2.22T + 2T^{2} \) |
| 3 | \( 1 + 1.47T + 3T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 13 | \( 1 + 3.92T + 13T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 23 | \( 1 - 0.113T + 23T^{2} \) |
| 29 | \( 1 - 1.08T + 29T^{2} \) |
| 31 | \( 1 + 4.17T + 31T^{2} \) |
| 37 | \( 1 + 5.75T + 37T^{2} \) |
| 41 | \( 1 - 4.26T + 41T^{2} \) |
| 43 | \( 1 - 3.62T + 43T^{2} \) |
| 47 | \( 1 + 6.39T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 0.883T + 59T^{2} \) |
| 61 | \( 1 - 3.77T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 - 4.28T + 71T^{2} \) |
| 73 | \( 1 + 1.92T + 73T^{2} \) |
| 79 | \( 1 + 7.81T + 79T^{2} \) |
| 83 | \( 1 + 3.33T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−9.741063629422071909327386162703, −8.857048145900172957948912625969, −7.16701825764086779906719293890, −6.65186201697669140569987061136, −5.78538816656008188925923333597, −5.37753319588835800476599691347, −4.33510243722719969775879883865, −3.19789646954358869850323480873, −2.47822128914369870164110130036, 0,
2.47822128914369870164110130036, 3.19789646954358869850323480873, 4.33510243722719969775879883865, 5.37753319588835800476599691347, 5.78538816656008188925923333597, 6.65186201697669140569987061136, 7.16701825764086779906719293890, 8.857048145900172957948912625969, 9.741063629422071909327386162703