| L(s) = 1 | − 2·3-s + 3·9-s − 4·11-s + 8·17-s − 8·19-s − 10·25-s − 4·27-s + 8·33-s + 16·41-s − 16·43-s + 49-s − 16·51-s + 16·57-s + 8·67-s − 4·73-s + 20·75-s + 5·81-s + 24·83-s + 12·97-s − 12·99-s + 12·107-s + 36·113-s − 10·121-s − 32·123-s + 127-s + 32·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 1.20·11-s + 1.94·17-s − 1.83·19-s − 2·25-s − 0.769·27-s + 1.39·33-s + 2.49·41-s − 2.43·43-s + 1/7·49-s − 2.24·51-s + 2.11·57-s + 0.977·67-s − 0.468·73-s + 2.30·75-s + 5/9·81-s + 2.63·83-s + 1.21·97-s − 1.20·99-s + 1.16·107-s + 3.38·113-s − 0.909·121-s − 2.88·123-s + 0.0887·127-s + 2.81·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−7.974616483504521309502852784131, −7.48026155837481865360444931206, −7.18271968465350908028898872220, −6.35375124938765540482595553493, −6.18939008741721547041883154292, −5.62712164599905299138472285079, −5.47671830652550627986429066605, −4.61868173456871870720539586920, −4.59244879976712680478557977307, −3.60350379570636772762568936671, −3.43433897571752185423069296584, −2.31369731751646119162884175837, −1.99001705800767037924689990798, −0.916495509655639876111635303243, 0,
0.916495509655639876111635303243, 1.99001705800767037924689990798, 2.31369731751646119162884175837, 3.43433897571752185423069296584, 3.60350379570636772762568936671, 4.59244879976712680478557977307, 4.61868173456871870720539586920, 5.47671830652550627986429066605, 5.62712164599905299138472285079, 6.18939008741721547041883154292, 6.35375124938765540482595553493, 7.18271968465350908028898872220, 7.48026155837481865360444931206, 7.974616483504521309502852784131
