| L(s) = 1 | − 3-s + 6·5-s + 2·7-s + 3·9-s − 9·11-s − 6·15-s − 4·17-s − 11·19-s − 2·21-s − 4·23-s + 9·25-s − 2·27-s − 3·29-s − 16·31-s + 9·33-s + 12·35-s − 4·37-s + 10·41-s + 43-s + 18·45-s + 32·47-s + 49-s + 4·51-s − 54·55-s + 11·57-s + 4·59-s + 6·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 2.68·5-s + 0.755·7-s + 9-s − 2.71·11-s − 1.54·15-s − 0.970·17-s − 2.52·19-s − 0.436·21-s − 0.834·23-s + 9/5·25-s − 0.384·27-s − 0.557·29-s − 2.87·31-s + 1.56·33-s + 2.02·35-s − 0.657·37-s + 1.56·41-s + 0.152·43-s + 2.68·45-s + 4.66·47-s + 1/7·49-s + 0.560·51-s − 7.28·55-s + 1.45·57-s + 0.520·59-s + 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.554051670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.554051670\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + T - 2 T^{2} - p T^{3} - p T^{4} - p^{2} T^{5} - 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 9 T + 42 T^{2} + 153 T^{3} + 509 T^{4} + 153 p T^{5} + 42 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 4 T - 9 T^{2} - 36 T^{3} + 64 T^{4} - 36 p T^{5} - 9 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 11 T + 56 T^{2} + 297 T^{3} + 1565 T^{4} + 297 p T^{5} + 56 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 3 T - 48 T^{2} - 3 T^{3} + 2147 T^{4} - 3 p T^{5} - 48 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 65 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T - 10 T^{2} - 192 T^{3} - 1285 T^{4} - 192 p T^{5} - 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 10 T + 6 T^{2} - 120 T^{3} + 3055 T^{4} - 120 p T^{5} + 6 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - T - 56 T^{2} + 29 T^{3} + 1357 T^{4} + 29 p T^{5} - 56 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 16 T + 145 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4 T - 93 T^{2} + 36 T^{3} + 7456 T^{4} + 36 p T^{5} - 93 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 24 T + 253 T^{2} - 24 p T^{3} + p^{2} T^{4} )( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $C_2^2$ | \( ( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 2 T - 22 T^{2} - 232 T^{3} - 4649 T^{4} - 232 p T^{5} - 22 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $D_{4}$ | \( ( 1 + 16 T + 217 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 21 T + 182 T^{2} + 1701 T^{3} + 19911 T^{4} + 1701 p T^{5} + 182 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 23 T + 232 T^{2} - 2369 T^{3} + 27487 T^{4} - 2369 p T^{5} + 232 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.093613534311314808538507129134, −7.998800979577138175609824982962, −7.928344581250233956817722093157, −7.38596674438009953330486377799, −7.25038928674439170479713036974, −7.17992736912262435092148485977, −6.77196669989586514359521555106, −6.26535634696070366813830564331, −6.15004011573173469883792758699, −5.85481695330638953558994276287, −5.65703401036892100952047137196, −5.56754909329035572350907897989, −5.42725490490203668547434590180, −4.97625393284363821588813098299, −4.58464912842067791282364016506, −4.38344948311692887120068685202, −4.08775994471865909359428653116, −3.84243154353666024426038462535, −3.24183421362289111243489981232, −2.45436803452976693604073378195, −2.42955462695472090292290956267, −2.15891319285075305321793199793, −1.87620254324778795540896787179, −1.70663876683441375589242825840, −0.46584609507010885795392899502,
0.46584609507010885795392899502, 1.70663876683441375589242825840, 1.87620254324778795540896787179, 2.15891319285075305321793199793, 2.42955462695472090292290956267, 2.45436803452976693604073378195, 3.24183421362289111243489981232, 3.84243154353666024426038462535, 4.08775994471865909359428653116, 4.38344948311692887120068685202, 4.58464912842067791282364016506, 4.97625393284363821588813098299, 5.42725490490203668547434590180, 5.56754909329035572350907897989, 5.65703401036892100952047137196, 5.85481695330638953558994276287, 6.15004011573173469883792758699, 6.26535634696070366813830564331, 6.77196669989586514359521555106, 7.17992736912262435092148485977, 7.25038928674439170479713036974, 7.38596674438009953330486377799, 7.928344581250233956817722093157, 7.998800979577138175609824982962, 8.093613534311314808538507129134
