| L(s) = 1 | + 2-s + 2·3-s − 4-s + 2·6-s − 3·8-s + 9-s − 4·11-s − 2·12-s − 2·13-s − 16-s + 18-s − 4·22-s + 12·23-s − 6·24-s + 25-s − 2·26-s − 4·27-s + 5·32-s − 8·33-s − 36-s − 4·37-s − 4·39-s + 4·44-s + 12·46-s − 8·47-s − 2·48-s + 2·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.816·6-s − 1.06·8-s + 1/3·9-s − 1.20·11-s − 0.577·12-s − 0.554·13-s − 1/4·16-s + 0.235·18-s − 0.852·22-s + 2.50·23-s − 1.22·24-s + 1/5·25-s − 0.392·26-s − 0.769·27-s + 0.883·32-s − 1.39·33-s − 1/6·36-s − 0.657·37-s − 0.640·39-s + 0.603·44-s + 1.76·46-s − 1.16·47-s − 0.288·48-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 608400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.050663651453834316178513676641, −7.893612143808235569104846929392, −7.36945570129601409328488904448, −6.86847438549507433409027285363, −6.35831289645710659676663930685, −5.66957165960173077695586779200, −5.26806860730181407315208823154, −4.76761858373058400551305930705, −4.59731852050444182415870036677, −3.51994456797006266949922249859, −3.43626038544626492191293151701, −2.65504338827892484384496886087, −2.53409816778618263405222233425, −1.34491844974665976517057296331, 0,
1.34491844974665976517057296331, 2.53409816778618263405222233425, 2.65504338827892484384496886087, 3.43626038544626492191293151701, 3.51994456797006266949922249859, 4.59731852050444182415870036677, 4.76761858373058400551305930705, 5.26806860730181407315208823154, 5.66957165960173077695586779200, 6.35831289645710659676663930685, 6.86847438549507433409027285363, 7.36945570129601409328488904448, 7.893612143808235569104846929392, 8.050663651453834316178513676641
