| L(s) = 1 | − 5·9-s − 6·11-s − 14·19-s + 6·29-s − 16·31-s + 18·41-s − 13·49-s − 6·59-s − 22·61-s + 6·71-s − 32·79-s + 9·81-s + 30·89-s + 30·99-s + 18·101-s − 8·109-s + 31·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | − 5/3·9-s − 1.80·11-s − 3.21·19-s + 1.11·29-s − 2.87·31-s + 2.81·41-s − 1.85·49-s − 0.781·59-s − 2.81·61-s + 0.712·71-s − 3.60·79-s + 81-s + 3.17·89-s + 3.01·99-s + 1.79·101-s − 0.766·109-s + 2.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1169473526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1169473526\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 25 T^{2} + 336 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 25 T^{2} - 744 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 35 T^{2} - 624 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 85 T^{2} + 2736 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 145 T^{2} + 11616 T^{4} + 145 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.82932768186750260362354614040, −6.78768150669893101422966502028, −6.18484676600272839631296213837, −6.12149317775390873943957847124, −6.09571745741769660715039390942, −5.84293606616738709292195646779, −5.79306900546594112274890482786, −5.29258290183418749009594707175, −4.95285301094020354681251178557, −4.90976243768591211394641310509, −4.87985257266244360694493807909, −4.36167341590215850690254994013, −4.18803156068264208028469364552, −3.84004998829276107969003638542, −3.81711933791940795984021518520, −3.26383747401934582499251188272, −2.92200621383550730704894561856, −2.78971946111275467648817873811, −2.78655613054410225162071174764, −2.24071121244869294372185674253, −1.95404237055168856822359118301, −1.87258606616503860171160156213, −1.33959680066744382447069439720, −0.53978687998732208114433326842, −0.10555060916408512927287111008,
0.10555060916408512927287111008, 0.53978687998732208114433326842, 1.33959680066744382447069439720, 1.87258606616503860171160156213, 1.95404237055168856822359118301, 2.24071121244869294372185674253, 2.78655613054410225162071174764, 2.78971946111275467648817873811, 2.92200621383550730704894561856, 3.26383747401934582499251188272, 3.81711933791940795984021518520, 3.84004998829276107969003638542, 4.18803156068264208028469364552, 4.36167341590215850690254994013, 4.87985257266244360694493807909, 4.90976243768591211394641310509, 4.95285301094020354681251178557, 5.29258290183418749009594707175, 5.79306900546594112274890482786, 5.84293606616738709292195646779, 6.09571745741769660715039390942, 6.12149317775390873943957847124, 6.18484676600272839631296213837, 6.78768150669893101422966502028, 6.82932768186750260362354614040
