| L(s) = 1 | − 9·5-s + 7-s − 36·11-s + 5·13-s − 2·19-s + 99·23-s + 22·25-s + 63·29-s + 7·31-s − 9·35-s − 64·37-s + 18·41-s + 46·43-s − 81·47-s + 24·49-s + 324·55-s + 126·59-s + 41·61-s − 45·65-s − 116·67-s + 86·73-s − 36·77-s − 83·79-s − 81·83-s + 5·91-s + 18·95-s − 196·97-s + ⋯ |
| L(s) = 1 | − 9/5·5-s + 1/7·7-s − 3.27·11-s + 5/13·13-s − 0.105·19-s + 4.30·23-s + 0.879·25-s + 2.17·29-s + 7/31·31-s − 0.257·35-s − 1.72·37-s + 0.439·41-s + 1.06·43-s − 1.72·47-s + 0.489·49-s + 5.89·55-s + 2.13·59-s + 0.672·61-s − 0.692·65-s − 1.73·67-s + 1.17·73-s − 0.467·77-s − 1.05·79-s − 0.975·83-s + 5/91·91-s + 0.189·95-s − 2.02·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5822451738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5822451738\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 9 T + 59 T^{2} + 288 T^{3} + 1074 T^{4} + 288 p^{2} T^{5} + 59 p^{4} T^{6} + 9 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - T - 23 T^{2} + 74 T^{3} - 1874 T^{4} + 74 p^{2} T^{5} - 23 p^{4} T^{6} - p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 36 T + 683 T^{2} + 9036 T^{3} + 100632 T^{4} + 9036 p^{2} T^{5} + 683 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 5 T - 245 T^{2} + 340 T^{3} + 40114 T^{4} + 340 p^{2} T^{5} - 245 p^{4} T^{6} - 5 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 769 T^{2} + 298176 T^{4} - 769 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + T + 648 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 99 T + 5117 T^{2} - 183150 T^{3} + 4870902 T^{4} - 183150 p^{2} T^{5} + 5117 p^{4} T^{6} - 99 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 63 T + 2123 T^{2} - 50400 T^{3} + 1045362 T^{4} - 50400 p^{2} T^{5} + 2123 p^{4} T^{6} - 63 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 7 T - 1217 T^{2} + 4592 T^{3} + 632146 T^{4} + 4592 p^{2} T^{5} - 1217 p^{4} T^{6} - 7 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 32 T + 1806 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 18 T + 1913 T^{2} - 32490 T^{3} + 613812 T^{4} - 32490 p^{2} T^{5} + 1913 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 23 T - 1320 T^{2} - 23 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 81 T + 6929 T^{2} + 384102 T^{3} + 22437966 T^{4} + 384102 p^{2} T^{5} + 6929 p^{4} T^{6} + 81 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 7204 T^{2} + 26018214 T^{4} - 7204 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 126 T + 11993 T^{2} - 844326 T^{3} + 51207492 T^{4} - 844326 p^{2} T^{5} + 11993 p^{4} T^{6} - 126 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 41 T - 5513 T^{2} + 10168 T^{3} + 31652794 T^{4} + 10168 p^{2} T^{5} - 5513 p^{4} T^{6} - 41 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 116 T + 3787 T^{2} + 80156 T^{3} + 12934456 T^{4} + 80156 p^{2} T^{5} + 3787 p^{4} T^{6} + 116 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 18616 T^{2} + 137194926 T^{4} - 18616 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 43 T + 10452 T^{2} - 43 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 83 T - 5459 T^{2} - 11122 T^{3} + 70528774 T^{4} - 11122 p^{2} T^{5} - 5459 p^{4} T^{6} + 83 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 81 T + 16289 T^{2} + 1142262 T^{3} + 166474326 T^{4} + 1142262 p^{2} T^{5} + 16289 p^{4} T^{6} + 81 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6916 T^{2} + 69013446 T^{4} - 6916 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 196 T + 10291 T^{2} + 1824172 T^{3} + 341030200 T^{4} + 1824172 p^{2} T^{5} + 10291 p^{4} T^{6} + 196 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−7.954462800412299223774631351442, −7.57200012508979880097696628044, −7.40751368444858164295824442027, −7.23951161023343908444810695886, −6.83578039756376643305952054491, −6.76275454506900782842984875629, −6.63749517195916949481865318531, −6.04428966960453023634890292319, −5.57447918962639702510055295503, −5.54443163501940799332103508261, −5.15614356859366847743402684494, −5.06038585530983970540392950505, −4.83767001297821086411037205403, −4.49869597471401759577878814414, −4.13787836068312638641152561030, −4.04758530054420438427999191107, −3.24280240460918905607988370041, −3.22196477421003218479515465631, −2.98308603211654009245275865014, −2.78295399942099358176685099038, −2.36653335536625783726019126306, −1.87319335159163778490913242882, −0.954552652145386159259831923371, −0.942568605267820162937512428788, −0.20139520616062212223795453517,
0.20139520616062212223795453517, 0.942568605267820162937512428788, 0.954552652145386159259831923371, 1.87319335159163778490913242882, 2.36653335536625783726019126306, 2.78295399942099358176685099038, 2.98308603211654009245275865014, 3.22196477421003218479515465631, 3.24280240460918905607988370041, 4.04758530054420438427999191107, 4.13787836068312638641152561030, 4.49869597471401759577878814414, 4.83767001297821086411037205403, 5.06038585530983970540392950505, 5.15614356859366847743402684494, 5.54443163501940799332103508261, 5.57447918962639702510055295503, 6.04428966960453023634890292319, 6.63749517195916949481865318531, 6.76275454506900782842984875629, 6.83578039756376643305952054491, 7.23951161023343908444810695886, 7.40751368444858164295824442027, 7.57200012508979880097696628044, 7.954462800412299223774631351442
