| L(s) = 1 | + 2-s − 4-s − 4·5-s − 3·8-s + 9-s − 4·10-s + 12·13-s − 16-s − 12·17-s + 18-s + 4·20-s + 2·25-s + 12·26-s + 4·29-s + 5·32-s − 12·34-s − 36-s − 20·37-s + 12·40-s − 4·41-s − 4·45-s − 14·49-s + 2·50-s − 12·52-s − 12·53-s + 4·58-s − 4·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.06·8-s + 1/3·9-s − 1.26·10-s + 3.32·13-s − 1/4·16-s − 2.91·17-s + 0.235·18-s + 0.894·20-s + 2/5·25-s + 2.35·26-s + 0.742·29-s + 0.883·32-s − 2.05·34-s − 1/6·36-s − 3.28·37-s + 1.89·40-s − 0.624·41-s − 0.596·45-s − 2·49-s + 0.282·50-s − 1.66·52-s − 1.64·53-s + 0.525·58-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.717867966311693207189859225084, −9.032116632714081125535639739797, −8.673289841767403383115719575464, −8.263513432862467471567669978548, −8.099644677533565214444467559710, −6.98236656892345640095319736227, −6.46787589567348650226556085189, −6.30774150521524867405980638808, −5.28029214515602867786662657712, −4.56991321080597356703354311172, −4.16838172205224943400021226417, −3.49609663069962101813276746331, −3.44365518362789223021999993571, −1.72203721066548968493435337820, 0,
1.72203721066548968493435337820, 3.44365518362789223021999993571, 3.49609663069962101813276746331, 4.16838172205224943400021226417, 4.56991321080597356703354311172, 5.28029214515602867786662657712, 6.30774150521524867405980638808, 6.46787589567348650226556085189, 6.98236656892345640095319736227, 8.099644677533565214444467559710, 8.263513432862467471567669978548, 8.673289841767403383115719575464, 9.032116632714081125535639739797, 9.717867966311693207189859225084
