| L(s) = 1 | + 3-s − 3·4-s + 9-s + 8·11-s − 3·12-s + 5·16-s − 4·17-s + 8·19-s + 25-s + 27-s + 4·29-s + 8·33-s − 3·36-s − 24·44-s + 5·48-s − 14·49-s − 4·51-s + 20·53-s + 8·57-s − 3·64-s + 24·67-s + 12·68-s + 75-s − 24·76-s + 81-s − 24·83-s + 4·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 3/2·4-s + 1/3·9-s + 2.41·11-s − 0.866·12-s + 5/4·16-s − 0.970·17-s + 1.83·19-s + 1/5·25-s + 0.192·27-s + 0.742·29-s + 1.39·33-s − 1/2·36-s − 3.61·44-s + 0.721·48-s − 2·49-s − 0.560·51-s + 2.74·53-s + 1.05·57-s − 3/8·64-s + 2.93·67-s + 1.45·68-s + 0.115·75-s − 2.75·76-s + 1/9·81-s − 2.63·83-s + 0.428·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.113713401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.113713401\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T ) \) |
| 31 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−8.554588868470699894187777077269, −8.171383858653182102218519209206, −7.48835909615996771691369301256, −6.89399677945660974091495871190, −6.76955885158912340897756950945, −6.12735884290641290480561231131, −5.46521203015847388109903840618, −5.04860090093668311608273372623, −4.44292527180552424855002162395, −4.12230566646888283690358404375, −3.63412818236731287070311550143, −3.24109761070285915353404768289, −2.33366107548066026326855143182, −1.38363062337439954492993809497, −0.832782540282442584365725306759,
0.832782540282442584365725306759, 1.38363062337439954492993809497, 2.33366107548066026326855143182, 3.24109761070285915353404768289, 3.63412818236731287070311550143, 4.12230566646888283690358404375, 4.44292527180552424855002162395, 5.04860090093668311608273372623, 5.46521203015847388109903840618, 6.12735884290641290480561231131, 6.76955885158912340897756950945, 6.89399677945660974091495871190, 7.48835909615996771691369301256, 8.171383858653182102218519209206, 8.554588868470699894187777077269
