| L(s) = 1 | − 2·2-s − 4-s + 4·7-s + 8·8-s − 2·11-s − 8·14-s − 7·16-s + 4·22-s − 16·23-s − 6·25-s − 4·28-s + 12·29-s − 14·32-s + 12·37-s + 2·44-s + 32·46-s + 9·49-s + 12·50-s − 12·53-s + 32·56-s − 24·58-s + 35·64-s − 8·67-s − 24·74-s − 8·77-s − 8·79-s − 16·88-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 1.51·7-s + 2.82·8-s − 0.603·11-s − 2.13·14-s − 7/4·16-s + 0.852·22-s − 3.33·23-s − 6/5·25-s − 0.755·28-s + 2.22·29-s − 2.47·32-s + 1.97·37-s + 0.301·44-s + 4.71·46-s + 9/7·49-s + 1.69·50-s − 1.64·53-s + 4.27·56-s − 3.15·58-s + 35/8·64-s − 0.977·67-s − 2.78·74-s − 0.911·77-s − 0.900·79-s − 1.70·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.389268452519734683124359977272, −8.092649848612792766954418771551, −7.60731726518247812897599595849, −7.55113610427119386648970831546, −6.55118736577760269645886929040, −5.83278296173294764308357143188, −5.64913594526162848677189952595, −4.65970646396455232760718697930, −4.52395871965258729630923756023, −4.28499925734034658438358048834, −3.43632280535116392331032608080, −2.26752106768048411048763527476, −1.82001999803537272093437549398, −1.03326922256959689494698211571, 0,
1.03326922256959689494698211571, 1.82001999803537272093437549398, 2.26752106768048411048763527476, 3.43632280535116392331032608080, 4.28499925734034658438358048834, 4.52395871965258729630923756023, 4.65970646396455232760718697930, 5.64913594526162848677189952595, 5.83278296173294764308357143188, 6.55118736577760269645886929040, 7.55113610427119386648970831546, 7.60731726518247812897599595849, 8.092649848612792766954418771551, 8.389268452519734683124359977272
