| L(s) = 1 | + 2·3-s − 4·5-s − 6·7-s + 2·9-s − 8·15-s − 12·21-s + 10·25-s + 6·27-s + 24·35-s − 32·41-s − 20·43-s − 8·45-s − 4·47-s + 18·49-s − 12·63-s + 12·67-s + 20·75-s + 24·79-s + 11·81-s − 20·83-s − 32·89-s + 8·101-s + 48·105-s + 24·109-s + 16·121-s − 64·123-s − 20·125-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s − 2.26·7-s + 2/3·9-s − 2.06·15-s − 2.61·21-s + 2·25-s + 1.15·27-s + 4.05·35-s − 4.99·41-s − 3.04·43-s − 1.19·45-s − 0.583·47-s + 18/7·49-s − 1.51·63-s + 1.46·67-s + 2.30·75-s + 2.70·79-s + 11/9·81-s − 2.19·83-s − 3.39·89-s + 0.796·101-s + 4.68·105-s + 2.29·109-s + 1.45·121-s − 5.77·123-s − 1.78·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1796185793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1796185793\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| good | 11 | $C_4\times C_2$ | \( 1 - 16 T^{2} + 126 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 286 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 5038 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_4$ | \( ( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 10 T + 106 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 7342 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 2998 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 9838 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 12638 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 10 T + 186 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 16 T + 222 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 17918 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.152504848560974395356514449974, −8.144072270068138987658949031568, −7.56823755887703292829274846459, −7.41859922603523780286884010778, −6.94296996869852914715587024927, −6.83449330724216762560194347890, −6.75684513321432908225229995446, −6.71868273291628556324694221350, −6.23569567507222654965781937079, −5.97703583517688998697097113948, −5.52696253954270562868968807154, −5.10765887403387151069171293952, −4.94404154571157515864879251935, −4.82396254806836367321165365139, −4.30005880121403821981608870605, −3.88963108707000860014831668196, −3.78819398545219147696692162072, −3.33415260326266015183406669701, −3.26502875099632618278479471571, −3.01126012207753158160651467193, −2.97806278520627561433182190339, −2.04547161864513691826590735239, −1.93037446463870524463219336394, −1.13345449806548433860708418554, −0.15540025662594980281334859618,
0.15540025662594980281334859618, 1.13345449806548433860708418554, 1.93037446463870524463219336394, 2.04547161864513691826590735239, 2.97806278520627561433182190339, 3.01126012207753158160651467193, 3.26502875099632618278479471571, 3.33415260326266015183406669701, 3.78819398545219147696692162072, 3.88963108707000860014831668196, 4.30005880121403821981608870605, 4.82396254806836367321165365139, 4.94404154571157515864879251935, 5.10765887403387151069171293952, 5.52696253954270562868968807154, 5.97703583517688998697097113948, 6.23569567507222654965781937079, 6.71868273291628556324694221350, 6.75684513321432908225229995446, 6.83449330724216762560194347890, 6.94296996869852914715587024927, 7.41859922603523780286884010778, 7.56823755887703292829274846459, 8.144072270068138987658949031568, 8.152504848560974395356514449974
