L(s) = 1 | + 2·3-s + 4-s + 9-s + 12·11-s + 2·12-s + 4·13-s + 8·17-s + 12·19-s − 4·23-s − 2·25-s − 2·27-s − 4·29-s + 24·33-s + 36-s + 18·37-s + 8·39-s − 10·43-s + 12·44-s − 10·49-s + 16·51-s + 4·52-s − 24·53-s + 24·57-s + 12·59-s − 64-s − 12·67-s + 8·68-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/2·4-s + 1/3·9-s + 3.61·11-s + 0.577·12-s + 1.10·13-s + 1.94·17-s + 2.75·19-s − 0.834·23-s − 2/5·25-s − 0.384·27-s − 0.742·29-s + 4.17·33-s + 1/6·36-s + 2.95·37-s + 1.28·39-s − 1.52·43-s + 1.80·44-s − 1.42·49-s + 2.24·51-s + 0.554·52-s − 3.29·53-s + 3.17·57-s + 1.56·59-s − 1/8·64-s − 1.46·67-s + 0.970·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.514727461\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.514727461\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
good | 7 | $C_2^3$ | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 12 T + 73 T^{2} - 300 T^{3} + 1032 T^{4} - 300 p T^{5} + 73 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 12 T + 82 T^{2} - 408 T^{3} + 1707 T^{4} - 408 p T^{5} + 82 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T - 31 T^{2} + 4 T^{3} + 1312 T^{4} + 4 p T^{5} - 31 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 4 T - 43 T^{2} + 4 T^{3} + 2176 T^{4} + 4 p T^{5} - 43 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 18 T + 193 T^{2} - 1530 T^{3} + 9852 T^{4} - 1530 p T^{5} + 193 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 78 T^{2} + 4403 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T + 37 T^{2} - 230 T^{3} - 2180 T^{4} - 230 p T^{5} + 37 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 66 T^{2} + 3155 T^{4} - 66 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T + 153 T^{2} - 1260 T^{3} + 10376 T^{4} - 1260 p T^{5} + 153 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 14 T^{2} - 3525 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 130 T^{2} + 984 T^{3} + 5451 T^{4} + 984 p T^{5} + 130 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 36 T + 678 T^{2} + 8856 T^{3} + 86147 T^{4} + 8856 p T^{5} + 678 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 14 T + 159 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 26006 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 222 T^{2} - 2088 T^{3} + 26627 T^{4} - 2088 p T^{5} + 222 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 12 T + 238 T^{2} + 2280 T^{3} + 31347 T^{4} + 2280 p T^{5} + 238 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.078373952676704552512322283425, −7.75357349541346647192320341784, −7.75338668018561427806800200379, −7.58993471480117470236079886203, −7.52100024155861497188637053801, −6.79082252273857290333450803593, −6.63623566718143719176609460135, −6.45520172121432949839163273474, −6.43490527312637545586264229066, −5.78617412331234509461318941352, −5.77407397996638689192526797988, −5.49456691732621386830599719431, −5.30852660327261309616696853991, −4.38025457321253419724202458380, −4.35518390664800572302136329605, −4.31498322252172755025692376795, −3.85457115966113509359783228067, −3.28280422320829006610625162403, −3.22727748026356694840297128846, −3.18842569997507225046348647153, −2.87138954784061684477933101437, −1.96770013224055879699421583161, −1.49669044634574517470979148698, −1.37082138207289870932682328220, −1.17735732094833667893933524257,
1.17735732094833667893933524257, 1.37082138207289870932682328220, 1.49669044634574517470979148698, 1.96770013224055879699421583161, 2.87138954784061684477933101437, 3.18842569997507225046348647153, 3.22727748026356694840297128846, 3.28280422320829006610625162403, 3.85457115966113509359783228067, 4.31498322252172755025692376795, 4.35518390664800572302136329605, 4.38025457321253419724202458380, 5.30852660327261309616696853991, 5.49456691732621386830599719431, 5.77407397996638689192526797988, 5.78617412331234509461318941352, 6.43490527312637545586264229066, 6.45520172121432949839163273474, 6.63623566718143719176609460135, 6.79082252273857290333450803593, 7.52100024155861497188637053801, 7.58993471480117470236079886203, 7.75338668018561427806800200379, 7.75357349541346647192320341784, 8.078373952676704552512322283425