| L(s) = 1 | + 3-s + 4-s − 2·7-s + 9-s + 12-s + 12·13-s + 16-s + 12·19-s − 2·21-s − 10·25-s + 27-s − 2·28-s − 4·31-s + 36-s + 20·37-s + 12·39-s + 16·43-s + 48-s + 3·49-s + 12·52-s + 12·57-s − 4·61-s − 2·63-s + 64-s − 8·67-s + 24·73-s − 10·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 0.755·7-s + 1/3·9-s + 0.288·12-s + 3.32·13-s + 1/4·16-s + 2.75·19-s − 0.436·21-s − 2·25-s + 0.192·27-s − 0.377·28-s − 0.718·31-s + 1/6·36-s + 3.28·37-s + 1.92·39-s + 2.43·43-s + 0.144·48-s + 3/7·49-s + 1.66·52-s + 1.58·57-s − 0.512·61-s − 0.251·63-s + 1/8·64-s − 0.977·67-s + 2.80·73-s − 1.15·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.573323326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.573323326\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−8.197346169778720895812897743279, −7.79089756844307556875120056782, −7.76942870316508074151045948135, −7.05017713754858032134757992630, −6.50358738357793944503631024403, −6.04478799301580802001339575092, −5.61371504482077087781349905589, −5.61053425475447892671224791397, −4.27334010758862108959056723897, −3.88277706681863956744577131696, −3.66659975798484481234060481751, −2.90909027996993607554406630510, −2.59024020521701587582121097787, −1.28532701621031142017445445957, −1.19027033408509088196634720414,
1.19027033408509088196634720414, 1.28532701621031142017445445957, 2.59024020521701587582121097787, 2.90909027996993607554406630510, 3.66659975798484481234060481751, 3.88277706681863956744577131696, 4.27334010758862108959056723897, 5.61053425475447892671224791397, 5.61371504482077087781349905589, 6.04478799301580802001339575092, 6.50358738357793944503631024403, 7.05017713754858032134757992630, 7.76942870316508074151045948135, 7.79089756844307556875120056782, 8.197346169778720895812897743279
