L(s) = 1 | − 2-s + 4-s + 6·7-s − 8-s − 6·14-s + 16-s − 2·25-s + 6·28-s − 2·29-s − 32-s − 4·41-s + 12·43-s + 14·49-s + 2·50-s + 4·53-s − 6·56-s + 2·58-s − 18·59-s + 6·61-s + 64-s + 22·71-s − 8·73-s + 4·82-s − 12·86-s + 14·89-s − 14·98-s − 2·100-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 2.26·7-s − 0.353·8-s − 1.60·14-s + 1/4·16-s − 2/5·25-s + 1.13·28-s − 0.371·29-s − 0.176·32-s − 0.624·41-s + 1.82·43-s + 2·49-s + 0.282·50-s + 0.549·53-s − 0.801·56-s + 0.262·58-s − 2.34·59-s + 0.768·61-s + 1/8·64-s + 2.61·71-s − 0.936·73-s + 0.441·82-s − 1.29·86-s + 1.48·89-s − 1.41·98-s − 1/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.831475579\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.831475579\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.575123820377966073009279804826, −7.974368155186343236715154734041, −7.84318373580892462855177977780, −7.39601444899731346360506008224, −6.93489488273043333121826416768, −6.18735702415022230220408204320, −5.84803946438893492228169499439, −5.10443447249364929799586547875, −4.91969828026160153423636034923, −4.26088306761105964994064629262, −3.72398950078021840279515626444, −2.87373342424849861647434678354, −2.06261859027072925778772451046, −1.73244853497995526329703158669, −0.855191706997318707696698986569,
0.855191706997318707696698986569, 1.73244853497995526329703158669, 2.06261859027072925778772451046, 2.87373342424849861647434678354, 3.72398950078021840279515626444, 4.26088306761105964994064629262, 4.91969828026160153423636034923, 5.10443447249364929799586547875, 5.84803946438893492228169499439, 6.18735702415022230220408204320, 6.93489488273043333121826416768, 7.39601444899731346360506008224, 7.84318373580892462855177977780, 7.974368155186343236715154734041, 8.575123820377966073009279804826