| L(s) = 1 | + 3-s − 3·4-s − 4·7-s + 9-s − 3·12-s + 5·16-s + 2·19-s − 4·21-s + 25-s + 27-s + 12·28-s − 3·36-s + 8·37-s − 4·43-s + 5·48-s − 2·49-s + 2·57-s + 4·61-s − 4·63-s − 3·64-s − 16·67-s + 28·73-s + 75-s − 6·76-s + 16·79-s + 81-s + 12·84-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 3/2·4-s − 1.51·7-s + 1/3·9-s − 0.866·12-s + 5/4·16-s + 0.458·19-s − 0.872·21-s + 1/5·25-s + 0.192·27-s + 2.26·28-s − 1/2·36-s + 1.31·37-s − 0.609·43-s + 0.721·48-s − 2/7·49-s + 0.264·57-s + 0.512·61-s − 0.503·63-s − 3/8·64-s − 1.95·67-s + 3.27·73-s + 0.115·75-s − 0.688·76-s + 1.80·79-s + 1/9·81-s + 1.30·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−8.836488963226532523388598253810, −8.311172749518175221259006108214, −7.944956574862892187546933534941, −7.44218328127442158412477604159, −6.69343741358824607018216997419, −6.43277558403273238107542389585, −5.81033642723526554045616411588, −5.08269165937541766478904797647, −4.81376783679940109514525317705, −3.93065235513561826916389859535, −3.73246630802271223857044732290, −3.06388291591923378327344798065, −2.45875404104195454497297562929, −1.15112547813740378682858966686, 0,
1.15112547813740378682858966686, 2.45875404104195454497297562929, 3.06388291591923378327344798065, 3.73246630802271223857044732290, 3.93065235513561826916389859535, 4.81376783679940109514525317705, 5.08269165937541766478904797647, 5.81033642723526554045616411588, 6.43277558403273238107542389585, 6.69343741358824607018216997419, 7.44218328127442158412477604159, 7.944956574862892187546933534941, 8.311172749518175221259006108214, 8.836488963226532523388598253810
