| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 12-s + 13-s − 14-s + 16-s + 17-s − 21-s + 24-s + 26-s − 27-s − 28-s − 2·31-s + 32-s + 34-s − 37-s + 39-s − 42-s + 43-s − 47-s + 48-s + 51-s + 52-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 12-s + 13-s − 14-s + 16-s + 17-s − 21-s + 24-s + 26-s − 27-s − 28-s − 2·31-s + 32-s + 34-s − 37-s + 39-s − 42-s + 43-s − 47-s + 48-s + 51-s + 52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.937828828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.937828828\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 + T )^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.062769504736937049464164672622, −8.198090997618067693422399203192, −7.49821455172757922165757387016, −6.68427215274052095903782513391, −5.89251161240254398719647704598, −5.24989238322250425751791968434, −3.83402563366697398815379581654, −3.51510687844275888770154519992, −2.74763920454551028347239033883, −1.64561752755130951743566934461,
1.64561752755130951743566934461, 2.74763920454551028347239033883, 3.51510687844275888770154519992, 3.83402563366697398815379581654, 5.24989238322250425751791968434, 5.89251161240254398719647704598, 6.68427215274052095903782513391, 7.49821455172757922165757387016, 8.198090997618067693422399203192, 9.062769504736937049464164672622
