| L(s) = 1 | − 3.40·3-s − 1.22·5-s + 7-s + 8.57·9-s + 4.94·11-s + 0.454·13-s + 4.17·15-s + 7.25·17-s − 19-s − 3.40·21-s + 1.54·23-s − 3.49·25-s − 18.9·27-s + 4.03·29-s − 8.35·31-s − 16.8·33-s − 1.22·35-s − 8.03·37-s − 1.54·39-s − 2.77·41-s − 6.03·43-s − 10.5·45-s + 0.597·47-s + 49-s − 24.7·51-s + 8.20·53-s − 6.07·55-s + ⋯ |
| L(s) = 1 | − 1.96·3-s − 0.548·5-s + 0.377·7-s + 2.85·9-s + 1.49·11-s + 0.125·13-s + 1.07·15-s + 1.76·17-s − 0.229·19-s − 0.742·21-s + 0.322·23-s − 0.698·25-s − 3.65·27-s + 0.748·29-s − 1.49·31-s − 2.93·33-s − 0.207·35-s − 1.32·37-s − 0.247·39-s − 0.433·41-s − 0.919·43-s − 1.56·45-s + 0.0871·47-s + 0.142·49-s − 3.45·51-s + 1.12·53-s − 0.818·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 3.40T + 3T^{2} \) |
| 5 | \( 1 + 1.22T + 5T^{2} \) |
| 11 | \( 1 - 4.94T + 11T^{2} \) |
| 13 | \( 1 - 0.454T + 13T^{2} \) |
| 17 | \( 1 - 7.25T + 17T^{2} \) |
| 23 | \( 1 - 1.54T + 23T^{2} \) |
| 29 | \( 1 - 4.03T + 29T^{2} \) |
| 31 | \( 1 + 8.35T + 31T^{2} \) |
| 37 | \( 1 + 8.03T + 37T^{2} \) |
| 41 | \( 1 + 2.77T + 41T^{2} \) |
| 43 | \( 1 + 6.03T + 43T^{2} \) |
| 47 | \( 1 - 0.597T + 47T^{2} \) |
| 53 | \( 1 - 8.20T + 53T^{2} \) |
| 59 | \( 1 + 2.31T + 59T^{2} \) |
| 61 | \( 1 + 0.143T + 61T^{2} \) |
| 67 | \( 1 + 1.25T + 67T^{2} \) |
| 71 | \( 1 - 0.948T + 71T^{2} \) |
| 73 | \( 1 + 7.89T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 7.09T + 83T^{2} \) |
| 89 | \( 1 - 2.94T + 89T^{2} \) |
| 97 | \( 1 + 3.85T + 97T^{2} \) |
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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
−7.11112182115085410648606980086, −6.82815965780422193161091147174, −5.87926827515724052261598016820, −5.52001643745958183784302632548, −4.75295064691961968010898748900, −4.00095373329848451302148825250, −3.49561870694169322802113584692, −1.64258285855059893799413357882, −1.14871470525869076832307503918, 0,
1.14871470525869076832307503918, 1.64258285855059893799413357882, 3.49561870694169322802113584692, 4.00095373329848451302148825250, 4.75295064691961968010898748900, 5.52001643745958183784302632548, 5.87926827515724052261598016820, 6.82815965780422193161091147174, 7.11112182115085410648606980086
