| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8·7-s − 8-s + 9-s + 2·10-s + 8·14-s + 16-s − 18-s − 8·19-s − 2·20-s + 3·25-s − 8·28-s − 32-s + 16·35-s + 36-s + 4·37-s + 8·38-s + 2·40-s − 8·43-s − 2·45-s + 34·49-s − 3·50-s − 12·53-s + 8·56-s − 8·63-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 3.02·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 2.13·14-s + 1/4·16-s − 0.235·18-s − 1.83·19-s − 0.447·20-s + 3/5·25-s − 1.51·28-s − 0.176·32-s + 2.70·35-s + 1/6·36-s + 0.657·37-s + 1.29·38-s + 0.316·40-s − 1.21·43-s − 0.298·45-s + 34/7·49-s − 0.424·50-s − 1.64·53-s + 1.06·56-s − 1.00·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 871200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 871200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.941231322257043910223245152113, −7.66751477528764076621092472724, −6.91345710014866780300605097423, −6.59687523488939699325560511774, −6.42217617652666865799421983967, −6.16029416330558685391350369510, −5.33434300745035533802295807658, −4.64771249748701799658409503507, −4.06327726820177801997023046782, −3.44709741096786719323216183645, −3.36585804145949552210873743484, −2.62774056304714643466775377230, −1.99977719272447350928607623660, −0.65952467849615073938622716771, 0,
0.65952467849615073938622716771, 1.99977719272447350928607623660, 2.62774056304714643466775377230, 3.36585804145949552210873743484, 3.44709741096786719323216183645, 4.06327726820177801997023046782, 4.64771249748701799658409503507, 5.33434300745035533802295807658, 6.16029416330558685391350369510, 6.42217617652666865799421983967, 6.59687523488939699325560511774, 6.91345710014866780300605097423, 7.66751477528764076621092472724, 7.941231322257043910223245152113
