| L(s) = 1 | + 3-s + 9-s − 8·19-s + 12·23-s − 10·25-s + 27-s − 12·29-s − 8·43-s − 24·47-s + 49-s + 12·53-s − 8·57-s + 16·67-s + 12·69-s − 12·71-s − 20·73-s − 10·75-s + 81-s − 12·87-s − 20·97-s + 24·101-s + 14·121-s + 127-s − 8·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.83·19-s + 2.50·23-s − 2·25-s + 0.192·27-s − 2.22·29-s − 1.21·43-s − 3.50·47-s + 1/7·49-s + 1.64·53-s − 1.05·57-s + 1.95·67-s + 1.44·69-s − 1.42·71-s − 2.34·73-s − 1.15·75-s + 1/9·81-s − 1.28·87-s − 2.03·97-s + 2.38·101-s + 1.27·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.353128748295719499474283851341, −8.300043091642042619777544931378, −7.61029397060529576114922058501, −6.99345965309954841855438264197, −6.93446262615281462668117301613, −6.11056271857714724112691707067, −5.74578744061641052444639897263, −5.03121360889898514265616765984, −4.63917458628061830491997962672, −3.89806512783406200646548904839, −3.53462971911322619523210724383, −2.86774228706863151676919072059, −2.04734454071268201559695039191, −1.58867103529813382887688476545, 0,
1.58867103529813382887688476545, 2.04734454071268201559695039191, 2.86774228706863151676919072059, 3.53462971911322619523210724383, 3.89806512783406200646548904839, 4.63917458628061830491997962672, 5.03121360889898514265616765984, 5.74578744061641052444639897263, 6.11056271857714724112691707067, 6.93446262615281462668117301613, 6.99345965309954841855438264197, 7.61029397060529576114922058501, 8.300043091642042619777544931378, 8.353128748295719499474283851341
