| L(s) = 1 | + 2-s + 4-s + 3·5-s − 7-s + 8-s + 3·10-s + 3·11-s − 13-s − 14-s + 16-s + 6·17-s − 4·19-s + 3·20-s + 3·22-s − 3·23-s + 4·25-s − 26-s − 28-s − 4·31-s + 32-s + 6·34-s − 3·35-s + 8·37-s − 4·38-s + 3·40-s + 9·41-s − 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.377·7-s + 0.353·8-s + 0.948·10-s + 0.904·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.670·20-s + 0.639·22-s − 0.625·23-s + 4/5·25-s − 0.196·26-s − 0.188·28-s − 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.507·35-s + 1.31·37-s − 0.648·38-s + 0.474·40-s + 1.40·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.184284328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.184284328\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.742709189690839321016951734240, −9.392128167223059007516612781579, −8.144128364167005586019818554778, −7.11997069390722482521983813574, −6.08938575984296370150186104292, −5.87700891989987107264649896461, −4.68923690112583534816030632465, −3.64559755494929929444330525032, −2.53003511413320787525010514230, −1.46320312717712285932160015720,
1.46320312717712285932160015720, 2.53003511413320787525010514230, 3.64559755494929929444330525032, 4.68923690112583534816030632465, 5.87700891989987107264649896461, 6.08938575984296370150186104292, 7.11997069390722482521983813574, 8.144128364167005586019818554778, 9.392128167223059007516612781579, 9.742709189690839321016951734240
