| L(s) = 1 | + 2·2-s + 4-s + 2·5-s + 3·7-s − 2·8-s + 4·10-s − 5·11-s + 6·14-s − 4·16-s − 3·17-s − 8·19-s + 2·20-s − 10·22-s − 4·23-s − 11·25-s + 3·28-s + 8·29-s − 2·32-s − 6·34-s + 6·35-s − 11·37-s − 16·38-s − 4·40-s + 6·43-s − 5·44-s − 8·46-s + 4·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 0.894·5-s + 1.13·7-s − 0.707·8-s + 1.26·10-s − 1.50·11-s + 1.60·14-s − 16-s − 0.727·17-s − 1.83·19-s + 0.447·20-s − 2.13·22-s − 0.834·23-s − 2.19·25-s + 0.566·28-s + 1.48·29-s − 0.353·32-s − 1.02·34-s + 1.01·35-s − 1.80·37-s − 2.59·38-s − 0.632·40-s + 0.914·43-s − 0.753·44-s − 1.17·46-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.125159550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.125159550\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| good | 5 | $D_{4}$ | \( ( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 3 T - 4 T^{2} + 3 T^{3} + 57 T^{4} + 3 p T^{5} - 4 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 5 T + 15 T^{3} + 229 T^{4} + 15 p T^{5} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 3 T + 2 T^{2} - 81 T^{3} - 393 T^{4} - 81 p T^{5} + 2 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 8 T + 23 T^{2} + 24 T^{3} + 104 T^{4} + 24 p T^{5} + 23 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T - 21 T^{2} - 36 T^{3} + 472 T^{4} - 36 p T^{5} - 21 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 8 T + 3 T^{2} - 24 T^{3} + 1024 T^{4} - 24 p T^{5} + 3 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 11 T + 20 T^{2} + 297 T^{3} + 4355 T^{4} + 297 p T^{5} + 20 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 35 T^{2} - 456 T^{4} + 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6 T - 46 T^{2} + 24 T^{3} + 3327 T^{4} + 24 p T^{5} - 46 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 + 25 T + 6 p T^{2} + 3825 T^{3} + 33085 T^{4} + 3825 p T^{5} + 6 p^{3} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 15 T + 50 T^{2} + 795 T^{3} + 13179 T^{4} + 795 p T^{5} + 50 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 9 T - 70 T^{2} - 153 T^{3} + 12885 T^{4} - 153 p T^{5} - 70 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 17 T + 78 T^{2} + 1173 T^{3} + 18961 T^{4} + 1173 p T^{5} + 78 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 13 T + 159 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 7 T + 141 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 47 T^{2} - 972 T^{3} - 11328 T^{4} - 972 p T^{5} + 47 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2 T - 74 T^{2} + 232 T^{3} - 3713 T^{4} + 232 p T^{5} - 74 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.44931710155077606701573428231, −7.39336877213513045991341390559, −7.04978983255978034289662667015, −6.53038239759625173192531350412, −6.45083983970373677243994309033, −6.32283785767617856387873362745, −5.83683665687215791952549977896, −5.78118850307618146031729945867, −5.77762306615379169465255777428, −5.27786471653958429536602100601, −5.02963971157197880670249197722, −4.91945304089873147997206047390, −4.63187556954007289097846632475, −4.35951409709449011408945432981, −4.13405019361891599841001095803, −3.83411503509831759783413542754, −3.78131676387272725834947081726, −3.15902558402276383427883431045, −2.94295667457664057063841697905, −2.36715786900079416652273695234, −2.25012810494293363524167302470, −2.23222473214889527123274306491, −1.69428987950209548240702949921, −1.24149652574381936140579056937, −0.20582211314878541047355333241,
0.20582211314878541047355333241, 1.24149652574381936140579056937, 1.69428987950209548240702949921, 2.23222473214889527123274306491, 2.25012810494293363524167302470, 2.36715786900079416652273695234, 2.94295667457664057063841697905, 3.15902558402276383427883431045, 3.78131676387272725834947081726, 3.83411503509831759783413542754, 4.13405019361891599841001095803, 4.35951409709449011408945432981, 4.63187556954007289097846632475, 4.91945304089873147997206047390, 5.02963971157197880670249197722, 5.27786471653958429536602100601, 5.77762306615379169465255777428, 5.78118850307618146031729945867, 5.83683665687215791952549977896, 6.32283785767617856387873362745, 6.45083983970373677243994309033, 6.53038239759625173192531350412, 7.04978983255978034289662667015, 7.39336877213513045991341390559, 7.44931710155077606701573428231
