| L(s) = 1 | + 9-s − 4·11-s + 4·16-s − 8·29-s − 28·31-s − 12·41-s + 11·49-s + 24·59-s + 26·61-s − 24·71-s − 12·79-s + 28·89-s − 4·99-s + 36·101-s + 44·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯ |
| L(s) = 1 | + 1/3·9-s − 1.20·11-s + 16-s − 1.48·29-s − 5.02·31-s − 1.87·41-s + 11/7·49-s + 3.12·59-s + 3.32·61-s − 2.84·71-s − 1.35·79-s + 2.96·89-s − 0.402·99-s + 3.58·101-s + 4.21·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1697903555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1697903555\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $C_2^2$ | \( ( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 85 T^{2} + 5376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 85 T^{2} + 2736 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 12 T + 73 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 79 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 167 T^{2} + p^{2} T^{4} ) \) |
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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
−7.26545674582985427115708487986, −7.03254375715097706848105471158, −6.85525967378697738372350314960, −6.65070581051895586765398137944, −5.97555833167327928234016489181, −5.90354986920880890578635481719, −5.88846466912045895027946812417, −5.58197557427453650140901561594, −5.35950304284896000498866983922, −5.17552345217781984027073613392, −4.97286447294961039211139624734, −4.54762172227435335766853372707, −4.53427654417164607186827999119, −3.88161273269324597235477527437, −3.65464600933144177021235355474, −3.50356819304089205918709775906, −3.41835739921571940215003882062, −3.40005673187505533327921984541, −2.43249984989716844857058722376, −2.21943833839312897957195844635, −2.19480513421728717475439883447, −1.96519662786247475715026195996, −1.23627388046903148292952137294, −1.03915961542936239534886886246, −0.098546290413917579213477070669,
0.098546290413917579213477070669, 1.03915961542936239534886886246, 1.23627388046903148292952137294, 1.96519662786247475715026195996, 2.19480513421728717475439883447, 2.21943833839312897957195844635, 2.43249984989716844857058722376, 3.40005673187505533327921984541, 3.41835739921571940215003882062, 3.50356819304089205918709775906, 3.65464600933144177021235355474, 3.88161273269324597235477527437, 4.53427654417164607186827999119, 4.54762172227435335766853372707, 4.97286447294961039211139624734, 5.17552345217781984027073613392, 5.35950304284896000498866983922, 5.58197557427453650140901561594, 5.88846466912045895027946812417, 5.90354986920880890578635481719, 5.97555833167327928234016489181, 6.65070581051895586765398137944, 6.85525967378697738372350314960, 7.03254375715097706848105471158, 7.26545674582985427115708487986
