| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s − 4·11-s + 13-s − 2·14-s + 16-s + 8·17-s − 6·19-s − 4·22-s + 6·23-s + 26-s − 2·28-s + 4·29-s + 32-s + 8·34-s + 2·37-s − 6·38-s + 2·41-s + 4·43-s − 4·44-s + 6·46-s − 3·49-s + 52-s − 10·53-s − 2·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 1.20·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 1.94·17-s − 1.37·19-s − 0.852·22-s + 1.25·23-s + 0.196·26-s − 0.377·28-s + 0.742·29-s + 0.176·32-s + 1.37·34-s + 0.328·37-s − 0.973·38-s + 0.312·41-s + 0.609·43-s − 0.603·44-s + 0.884·46-s − 3/7·49-s + 0.138·52-s − 1.37·53-s − 0.267·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.727225855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.727225855\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.79003551516086049978358436531, −7.53237715529879531138905824169, −6.32236866712103413082788478515, −6.12911584329025029627562112611, −5.10344794395496418386832407303, −4.63265733017116256296221359410, −3.40525780893678541212565970767, −3.09880228779188281769753009138, −2.09663474758288346911639767662, −0.77294040377296382231627851684,
0.77294040377296382231627851684, 2.09663474758288346911639767662, 3.09880228779188281769753009138, 3.40525780893678541212565970767, 4.63265733017116256296221359410, 5.10344794395496418386832407303, 6.12911584329025029627562112611, 6.32236866712103413082788478515, 7.53237715529879531138905824169, 7.79003551516086049978358436531
