| L(s) = 1 | + 3-s − 2·4-s + 9-s − 2·12-s − 2·13-s + 19-s − 25-s + 27-s + 3·31-s − 2·36-s − 4·37-s − 2·39-s − 10·43-s − 10·49-s + 4·52-s + 57-s − 2·61-s + 8·64-s + 8·67-s − 8·73-s − 75-s − 2·76-s + 81-s + 3·93-s − 14·97-s + 2·100-s − 2·108-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1/3·9-s − 0.577·12-s − 0.554·13-s + 0.229·19-s − 1/5·25-s + 0.192·27-s + 0.538·31-s − 1/3·36-s − 0.657·37-s − 0.320·39-s − 1.52·43-s − 1.42·49-s + 0.554·52-s + 0.132·57-s − 0.256·61-s + 64-s + 0.977·67-s − 0.936·73-s − 0.115·75-s − 0.229·76-s + 1/9·81-s + 0.311·93-s − 1.42·97-s + 1/5·100-s − 0.192·108-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 | $C_2$ | \( 1 + T^{2} \) |
| 19 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 93 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.633630525341493583638457503551, −8.395166941405474152027575516190, −7.914872422762183229290916967524, −7.41631768031240655787744141966, −6.80636502565617265499840869515, −6.47968958834526934246222304501, −5.69811719282712620007338569642, −5.05937629246028149837748720062, −4.81066540146337213509272971625, −4.17297521159942847561581740747, −3.61387392376109778270454534270, −3.01848660130097057822476697468, −2.27584932677099814084676102753, −1.39228421435216690840591183904, 0,
1.39228421435216690840591183904, 2.27584932677099814084676102753, 3.01848660130097057822476697468, 3.61387392376109778270454534270, 4.17297521159942847561581740747, 4.81066540146337213509272971625, 5.05937629246028149837748720062, 5.69811719282712620007338569642, 6.47968958834526934246222304501, 6.80636502565617265499840869515, 7.41631768031240655787744141966, 7.914872422762183229290916967524, 8.395166941405474152027575516190, 8.633630525341493583638457503551
