| L(s) = 1 | + 2·2-s + 3·4-s − 4·5-s + 2·8-s + 4·9-s − 8·10-s − 4·11-s + 14·13-s − 6·17-s + 8·18-s + 12·19-s − 12·20-s − 8·22-s + 6·25-s + 28·26-s − 14·29-s − 16·31-s − 6·32-s − 12·34-s + 12·36-s + 2·37-s + 24·38-s − 8·40-s + 6·41-s − 4·43-s − 12·44-s − 16·45-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.78·5-s + 0.707·8-s + 4/3·9-s − 2.52·10-s − 1.20·11-s + 3.88·13-s − 1.45·17-s + 1.88·18-s + 2.75·19-s − 2.68·20-s − 1.70·22-s + 6/5·25-s + 5.49·26-s − 2.59·29-s − 2.87·31-s − 1.06·32-s − 2.05·34-s + 2·36-s + 0.328·37-s + 3.89·38-s − 1.26·40-s + 0.937·41-s − 0.609·43-s − 1.80·44-s − 2.38·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.502986169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.502986169\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + T^{2} + p T^{3} - 3 T^{4} + p^{2} T^{5} + p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2^3$ | \( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 8 T^{2} + 8 T^{3} + 279 T^{4} + 8 p T^{5} - 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + T^{2} + 6 T^{3} + 324 T^{4} + 6 p T^{5} + p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 44 T^{2} + 1407 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 14 T + 97 T^{2} + 574 T^{3} + 3276 T^{4} + 574 p T^{5} + 97 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 76 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2 T + T^{2} + 142 T^{3} - 1508 T^{4} + 142 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6 T - 47 T^{2} - 6 T^{3} + 3732 T^{4} - 6 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T - 72 T^{2} + 8 T^{3} + 5207 T^{4} + 8 p T^{5} - 72 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 8 T^{2} + 216 T^{3} + 6519 T^{4} + 216 p T^{5} + 8 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 14 T + 33 T^{2} - 574 T^{3} + 10892 T^{4} - 574 p T^{5} + 33 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 - 116 T^{2} + 8967 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12 T - 2 T^{2} - 48 T^{3} + 6051 T^{4} - 48 p T^{5} - 2 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 10 T + 139 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 12 T + 176 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 152 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8 T - 2 T^{2} + 896 T^{3} - 9261 T^{4} + 896 p T^{5} - 2 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 16 T + 6 T^{2} + 896 T^{3} + 28259 T^{4} + 896 p T^{5} + 6 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.71492666965510169771977848986, −7.21948486032813490290390219184, −7.19874146076815876915081218209, −6.91609129150245368636605409471, −6.62119415739169307084011836113, −6.41814016821341507812484900466, −6.13005199983438260811236689557, −6.07136759738581580651873234626, −5.45316014604351275591986321742, −5.40562117390390258819417556553, −5.20864217871328387909301825506, −5.02562979209507215378980280201, −4.76694037878526961647975600959, −4.04288671912314537076766637627, −3.88041786186581089604105533248, −3.84223910874906252569008324359, −3.76579219571885375010852432725, −3.43001557290617064365149736583, −3.38808143717308652013818923618, −2.63327048722287997192053018669, −2.59338157833093095257779988036, −1.70954859475107915700543268333, −1.64863558962648945417496144642, −1.29767036431480818492215749011, −0.41931689561002645733451262767,
0.41931689561002645733451262767, 1.29767036431480818492215749011, 1.64863558962648945417496144642, 1.70954859475107915700543268333, 2.59338157833093095257779988036, 2.63327048722287997192053018669, 3.38808143717308652013818923618, 3.43001557290617064365149736583, 3.76579219571885375010852432725, 3.84223910874906252569008324359, 3.88041786186581089604105533248, 4.04288671912314537076766637627, 4.76694037878526961647975600959, 5.02562979209507215378980280201, 5.20864217871328387909301825506, 5.40562117390390258819417556553, 5.45316014604351275591986321742, 6.07136759738581580651873234626, 6.13005199983438260811236689557, 6.41814016821341507812484900466, 6.62119415739169307084011836113, 6.91609129150245368636605409471, 7.19874146076815876915081218209, 7.21948486032813490290390219184, 7.71492666965510169771977848986
