| L(s) = 1 | − 2·3-s − 2·5-s + 9-s + 4·15-s − 2·17-s − 6·19-s − 7·25-s + 4·27-s + 2·31-s − 2·45-s − 49-s + 4·51-s + 12·57-s − 2·59-s + 2·61-s − 8·67-s + 14·71-s + 14·73-s + 14·75-s − 16·79-s − 11·81-s + 4·85-s − 4·93-s + 12·95-s − 28·103-s − 24·107-s − 9·121-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 1/3·9-s + 1.03·15-s − 0.485·17-s − 1.37·19-s − 7/5·25-s + 0.769·27-s + 0.359·31-s − 0.298·45-s − 1/7·49-s + 0.560·51-s + 1.58·57-s − 0.260·59-s + 0.256·61-s − 0.977·67-s + 1.66·71-s + 1.63·73-s + 1.61·75-s − 1.80·79-s − 1.22·81-s + 0.433·85-s − 0.414·93-s + 1.23·95-s − 2.75·103-s − 2.32·107-s − 0.818·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2761605546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2761605546\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
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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
−8.192728714170956402737339872010, −7.82307053483805089684867818669, −7.31573250698658919178660427258, −6.70939417375030252544015898214, −6.46397349942181936038290244797, −6.07551337310252989429965529839, −5.42621860355612824043501757256, −5.16866250765144645597812064788, −4.43945814233442752744517093984, −4.10441092878549140339062261337, −3.74373412992296205235011459111, −2.85388904611726523196860864416, −2.27028111468721596056637040153, −1.40589042773190732408106565848, −0.27563242502208059945993766933,
0.27563242502208059945993766933, 1.40589042773190732408106565848, 2.27028111468721596056637040153, 2.85388904611726523196860864416, 3.74373412992296205235011459111, 4.10441092878549140339062261337, 4.43945814233442752744517093984, 5.16866250765144645597812064788, 5.42621860355612824043501757256, 6.07551337310252989429965529839, 6.46397349942181936038290244797, 6.70939417375030252544015898214, 7.31573250698658919178660427258, 7.82307053483805089684867818669, 8.192728714170956402737339872010
