| L(s) = 1 | + 2.61·3-s − 1.61·5-s + 7-s + 3.85·9-s − 3.85·11-s − 1.23·13-s − 4.23·15-s + 4·17-s − 19-s + 2.61·21-s − 0.763·23-s − 2.38·25-s + 2.23·27-s − 8.85·29-s + 4.47·31-s − 10.0·33-s − 1.61·35-s − 9.61·37-s − 3.23·39-s + 10.5·41-s + 10.5·43-s − 6.23·45-s − 6.32·47-s + 49-s + 10.4·51-s − 0.909·53-s + 6.23·55-s + ⋯ |
| L(s) = 1 | + 1.51·3-s − 0.723·5-s + 0.377·7-s + 1.28·9-s − 1.16·11-s − 0.342·13-s − 1.09·15-s + 0.970·17-s − 0.229·19-s + 0.571·21-s − 0.159·23-s − 0.476·25-s + 0.430·27-s − 1.64·29-s + 0.803·31-s − 1.75·33-s − 0.273·35-s − 1.58·37-s − 0.518·39-s + 1.64·41-s + 1.61·43-s − 0.929·45-s − 0.922·47-s + 0.142·49-s + 1.46·51-s − 0.124·53-s + 0.840·55-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 | \( 1 - 2.61T + 3T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 23 | \( 1 + 0.763T + 23T^{2} \) |
| 29 | \( 1 + 8.85T + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 + 9.61T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 6.32T + 47T^{2} \) |
| 53 | \( 1 + 0.909T + 53T^{2} \) |
| 59 | \( 1 - 5.61T + 59T^{2} \) |
| 61 | \( 1 - 0.618T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + 9.70T + 73T^{2} \) |
| 79 | \( 1 + 7.14T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 0.909T + 97T^{2} \) |
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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
−7.52749734924610530560299534050, −7.37220052089353120918117833587, −5.97817874108485176764495762855, −5.30367462380883532340562404213, −4.35904695122108166070886639254, −3.79645282387548265165958205876, −3.00687802871894665858967449691, −2.42069581190332953597918735084, −1.51028087440699572606135974385, 0,
1.51028087440699572606135974385, 2.42069581190332953597918735084, 3.00687802871894665858967449691, 3.79645282387548265165958205876, 4.35904695122108166070886639254, 5.30367462380883532340562404213, 5.97817874108485176764495762855, 7.37220052089353120918117833587, 7.52749734924610530560299534050
