| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 8·11-s + 12-s − 9·13-s + 16-s − 18-s + 8·22-s − 6·23-s − 24-s + 7·25-s + 9·26-s + 27-s − 32-s − 8·33-s + 36-s + 37-s − 9·39-s − 8·44-s + 6·46-s − 3·47-s + 48-s − 5·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 2.41·11-s + 0.288·12-s − 2.49·13-s + 1/4·16-s − 0.235·18-s + 1.70·22-s − 1.25·23-s − 0.204·24-s + 7/5·25-s + 1.76·26-s + 0.192·27-s − 0.176·32-s − 1.39·33-s + 1/6·36-s + 0.164·37-s − 1.44·39-s − 1.20·44-s + 0.884·46-s − 0.437·47-s + 0.144·48-s − 5/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40608 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40608 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 - T \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 43 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.933952688871811973548049902069, −9.616606498312106318196220124768, −9.058098056027617061982571293935, −8.232795075584774149964992173628, −7.912161152624164916129241422299, −7.62008136243859984430594798353, −7.05676912501080444674086331667, −6.41643659064863137620261645903, −5.43892818919932208442460849708, −5.02074460729049440231444501731, −4.49425375175569348825832238798, −3.17567839263068581671520029976, −2.62423488621626633121199031639, −2.10821115640498275456048893257, 0,
2.10821115640498275456048893257, 2.62423488621626633121199031639, 3.17567839263068581671520029976, 4.49425375175569348825832238798, 5.02074460729049440231444501731, 5.43892818919932208442460849708, 6.41643659064863137620261645903, 7.05676912501080444674086331667, 7.62008136243859984430594798353, 7.912161152624164916129241422299, 8.232795075584774149964992173628, 9.058098056027617061982571293935, 9.616606498312106318196220124768, 9.933952688871811973548049902069
