| L(s) = 1 | + 2·3-s − 2·4-s + 9-s − 6·11-s − 4·12-s − 8·13-s + 4·16-s − 25-s − 4·27-s − 12·33-s − 2·36-s + 4·37-s − 16·39-s + 12·44-s + 6·47-s + 8·48-s − 13·49-s + 16·52-s + 12·59-s − 2·61-s − 8·64-s − 12·71-s − 14·73-s − 2·75-s − 11·81-s − 24·83-s + 16·97-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s + 1/3·9-s − 1.80·11-s − 1.15·12-s − 2.21·13-s + 16-s − 1/5·25-s − 0.769·27-s − 2.08·33-s − 1/3·36-s + 0.657·37-s − 2.56·39-s + 1.80·44-s + 0.875·47-s + 1.15·48-s − 1.85·49-s + 2.21·52-s + 1.56·59-s − 0.256·61-s − 64-s − 1.42·71-s − 1.63·73-s − 0.230·75-s − 1.22·81-s − 2.63·83-s + 1.62·97-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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−9.834844837220933208096577473942, −9.238767264318381906771348588364, −8.867143248873807381249886076350, −8.105075898459616611364991061094, −7.968751144610877316457317025190, −7.43141403312000422638829977686, −6.97094335247891624980251070075, −5.68944505001593374643432676068, −5.51097905283736056328679977781, −4.62455059365688352928682233512, −4.41466206651142691425988858089, −3.23857552173567549926694025031, −2.80628438558787874241283090829, −2.10540195460897363050983106218, 0,
2.10540195460897363050983106218, 2.80628438558787874241283090829, 3.23857552173567549926694025031, 4.41466206651142691425988858089, 4.62455059365688352928682233512, 5.51097905283736056328679977781, 5.68944505001593374643432676068, 6.97094335247891624980251070075, 7.43141403312000422638829977686, 7.968751144610877316457317025190, 8.105075898459616611364991061094, 8.867143248873807381249886076350, 9.238767264318381906771348588364, 9.834844837220933208096577473942
