| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 4·11-s + 3·13-s − 14-s + 16-s + 7·17-s − 6·19-s − 4·22-s + 9·23-s − 3·26-s + 28-s + 3·29-s − 7·31-s − 32-s − 7·34-s + 10·37-s + 6·38-s − 41-s − 13·43-s + 4·44-s − 9·46-s − 2·47-s + 49-s + 3·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 1.20·11-s + 0.832·13-s − 0.267·14-s + 1/4·16-s + 1.69·17-s − 1.37·19-s − 0.852·22-s + 1.87·23-s − 0.588·26-s + 0.188·28-s + 0.557·29-s − 1.25·31-s − 0.176·32-s − 1.20·34-s + 1.64·37-s + 0.973·38-s − 0.156·41-s − 1.98·43-s + 0.603·44-s − 1.32·46-s − 0.291·47-s + 1/7·49-s + 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.704243311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.704243311\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.607085540051020140795133720691, −8.166299147268390496108552847747, −7.19823687288579167492296612273, −6.56332043882302736945426587055, −5.82878784320092650943298158719, −4.86467367893648732915427743342, −3.82347303189157010301354535705, −3.05653041823232478672605926362, −1.70846102076943940089378630816, −0.962218298664334543723658643839,
0.962218298664334543723658643839, 1.70846102076943940089378630816, 3.05653041823232478672605926362, 3.82347303189157010301354535705, 4.86467367893648732915427743342, 5.82878784320092650943298158719, 6.56332043882302736945426587055, 7.19823687288579167492296612273, 8.166299147268390496108552847747, 8.607085540051020140795133720691
