| L(s) = 1 | − 2·4-s + 7-s + 4·16-s + 7·19-s − 5·25-s − 2·28-s + 4·31-s − 11·37-s + 8·43-s − 6·49-s − 61-s − 8·64-s − 5·67-s + 7·73-s − 14·76-s + 17·79-s + 19·97-s + 10·100-s − 13·103-s − 2·109-s + 4·112-s + ⋯ |
| L(s) = 1 | − 4-s + 0.377·7-s + 16-s + 1.60·19-s − 25-s − 0.377·28-s + 0.718·31-s − 1.80·37-s + 1.21·43-s − 6/7·49-s − 0.128·61-s − 64-s − 0.610·67-s + 0.819·73-s − 1.60·76-s + 1.91·79-s + 1.92·97-s + 100-s − 1.28·103-s − 0.191·109-s + 0.377·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.469932292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.469932292\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.235455062110187587281357339471, −7.79905600784920878013549003135, −6.99386863016462421750119711316, −5.94590995620436746454346143991, −5.29061261743206461980587712737, −4.68510402165798343143294095308, −3.79824311881645584323392656481, −3.10103102437570631641825076290, −1.78574696324920418252589934325, −0.70162816863425891209859307802,
0.70162816863425891209859307802, 1.78574696324920418252589934325, 3.10103102437570631641825076290, 3.79824311881645584323392656481, 4.68510402165798343143294095308, 5.29061261743206461980587712737, 5.94590995620436746454346143991, 6.99386863016462421750119711316, 7.79905600784920878013549003135, 8.235455062110187587281357339471
