| L(s) = 1 | − 3-s + 2·5-s − 2·9-s − 11-s − 5·13-s − 2·15-s − 17-s + 6·19-s − 25-s + 5·27-s + 6·29-s − 4·31-s + 33-s − 8·37-s + 5·39-s + 6·41-s − 12·43-s − 4·45-s + 2·47-s + 51-s + 7·53-s − 2·55-s − 6·57-s − 12·59-s + 12·61-s − 10·65-s − 2·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 2/3·9-s − 0.301·11-s − 1.38·13-s − 0.516·15-s − 0.242·17-s + 1.37·19-s − 1/5·25-s + 0.962·27-s + 1.11·29-s − 0.718·31-s + 0.174·33-s − 1.31·37-s + 0.800·39-s + 0.937·41-s − 1.82·43-s − 0.596·45-s + 0.291·47-s + 0.140·51-s + 0.961·53-s − 0.269·55-s − 0.794·57-s − 1.56·59-s + 1.53·61-s − 1.24·65-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.127768006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.127768006\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.36007362019761, −13.97487800028248, −13.46973255066023, −12.97633002559076, −12.18078973023543, −11.97286772396647, −11.52329668718964, −10.71094438883568, −10.38230709315444, −9.799445696197015, −9.407997774666754, −8.803262105600037, −8.174605486740876, −7.536313823380067, −6.966710958666322, −6.479945965511296, −5.760980190438296, −5.253176004920758, −5.096408041224969, −4.254238434665316, −3.288003253595364, −2.761850373016039, −2.151182608816543, −1.364829314329812, −0.3803297313898806,
0.3803297313898806, 1.364829314329812, 2.151182608816543, 2.761850373016039, 3.288003253595364, 4.254238434665316, 5.096408041224969, 5.253176004920758, 5.760980190438296, 6.479945965511296, 6.966710958666322, 7.536313823380067, 8.174605486740876, 8.803262105600037, 9.407997774666754, 9.799445696197015, 10.38230709315444, 10.71094438883568, 11.52329668718964, 11.97286772396647, 12.18078973023543, 12.97633002559076, 13.46973255066023, 13.97487800028248, 14.36007362019761
