| L(s) = 1 | + 7-s + 9-s − 12·11-s + 4·13-s − 10·25-s + 16·31-s − 8·43-s + 24·47-s + 49-s − 20·61-s + 63-s + 16·67-s − 12·77-s + 81-s + 4·91-s − 12·99-s − 24·101-s + 16·103-s − 12·107-s − 12·113-s + 4·117-s + 86·121-s + 127-s + 131-s + 137-s + 139-s − 48·143-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1/3·9-s − 3.61·11-s + 1.10·13-s − 2·25-s + 2.87·31-s − 1.21·43-s + 3.50·47-s + 1/7·49-s − 2.56·61-s + 0.125·63-s + 1.95·67-s − 1.36·77-s + 1/9·81-s + 0.419·91-s − 1.20·99-s − 2.38·101-s + 1.57·103-s − 1.16·107-s − 1.12·113-s + 0.369·117-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.01·143-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.943837464504667779037827212581, −7.74489471466623660805940258686, −7.37585910878871251231103157573, −6.71214569469608680499201547060, −5.98540521798352690518559815387, −5.83454591780136502789057352294, −5.29620811558997278941087405158, −4.85164251678679590042275457849, −4.39050801784138122129668187119, −3.79628830333475548562993564774, −3.04530103925276638660028722430, −2.51752502799515976298852414682, −2.19768133803298656421701957429, −1.09057799706435300786341631543, 0,
1.09057799706435300786341631543, 2.19768133803298656421701957429, 2.51752502799515976298852414682, 3.04530103925276638660028722430, 3.79628830333475548562993564774, 4.39050801784138122129668187119, 4.85164251678679590042275457849, 5.29620811558997278941087405158, 5.83454591780136502789057352294, 5.98540521798352690518559815387, 6.71214569469608680499201547060, 7.37585910878871251231103157573, 7.74489471466623660805940258686, 7.943837464504667779037827212581
