| L(s) = 1 | + 1.80·2-s + 2.24·4-s + 7-s + 2.24·8-s − 1.80·13-s + 1.80·14-s + 1.80·16-s + 0.445·17-s − 0.445·19-s − 1.24·23-s + 25-s − 3.24·26-s + 2.24·28-s + 1.00·32-s + 0.801·34-s − 1.80·37-s − 0.801·38-s − 41-s − 0.445·43-s − 2.24·46-s + 1.80·47-s + 49-s + 1.80·50-s − 4.04·52-s + 2.24·56-s + 68-s − 3.24·74-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 2.24·4-s + 7-s + 2.24·8-s − 1.80·13-s + 1.80·14-s + 1.80·16-s + 0.445·17-s − 0.445·19-s − 1.24·23-s + 25-s − 3.24·26-s + 2.24·28-s + 1.00·32-s + 0.801·34-s − 1.80·37-s − 0.801·38-s − 41-s − 0.445·43-s − 2.24·46-s + 1.80·47-s + 49-s + 1.80·50-s − 4.04·52-s + 2.24·56-s + 68-s − 3.24·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.602451674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.602451674\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 1.80T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.80T + T^{2} \) |
| 17 | \( 1 - 0.445T + T^{2} \) |
| 19 | \( 1 + 0.445T + T^{2} \) |
| 23 | \( 1 + 1.24T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.80T + T^{2} \) |
| 43 | \( 1 + 0.445T + T^{2} \) |
| 47 | \( 1 - 1.80T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.80T + T^{2} \) |
| 97 | \( 1 - 1.24T + T^{2} \) |
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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
−8.983000295474648906364810549000, −7.976857762952515399555409853895, −7.28914788626878081825581172147, −6.66009108189749755568573142615, −5.59944926380598088895231367307, −5.05962152005061891764338708651, −4.48618270099382737373914540532, −3.59931104148218534627880623277, −2.54489260032120497780155444424, −1.83975269323596061561585091536,
1.83975269323596061561585091536, 2.54489260032120497780155444424, 3.59931104148218534627880623277, 4.48618270099382737373914540532, 5.05962152005061891764338708651, 5.59944926380598088895231367307, 6.66009108189749755568573142615, 7.28914788626878081825581172147, 7.976857762952515399555409853895, 8.983000295474648906364810549000
