| L(s) = 1 | + 2·5-s + 12·7-s − 6·9-s − 4·11-s + 2·13-s + 16·17-s + 12·19-s − 12·23-s − 25-s − 6·29-s + 8·31-s + 24·35-s − 24·37-s + 18·41-s − 8·43-s − 12·45-s − 8·47-s + 71·49-s + 16·53-s − 8·55-s − 12·59-s + 30·61-s − 72·63-s + 4·65-s − 16·67-s − 48·77-s + 27·81-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 4.53·7-s − 2·9-s − 1.20·11-s + 0.554·13-s + 3.88·17-s + 2.75·19-s − 2.50·23-s − 1/5·25-s − 1.11·29-s + 1.43·31-s + 4.05·35-s − 3.94·37-s + 2.81·41-s − 1.21·43-s − 1.78·45-s − 1.16·47-s + 71/7·49-s + 2.19·53-s − 1.07·55-s − 1.56·59-s + 3.84·61-s − 9.07·63-s + 0.496·65-s − 1.95·67-s − 5.47·77-s + 3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.615848566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.615848566\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 2 T + p T^{2} - 14 T^{3} + 16 T^{4} - 14 p T^{5} + p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 12 T + 73 T^{2} - 300 T^{3} + 912 T^{4} - 300 p T^{5} + 73 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T + 5 T^{2} - 20 T^{3} - 164 T^{4} - 20 p T^{5} + 5 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 97 T^{2} + 588 T^{3} + 2976 T^{4} + 588 p T^{5} + 97 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 6 T + 9 T^{2} - 6 p T^{3} - 1528 T^{4} - 6 p^{2} T^{5} + 9 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 8 T + 13 T^{2} + 88 T^{3} - 344 T^{4} + 88 p T^{5} + 13 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 2472 T^{3} + 16862 T^{4} + 2472 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 18 T + 181 T^{2} - 1314 T^{3} + 8076 T^{4} - 1314 p T^{5} + 181 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T + 65 T^{2} - 120 T^{3} - 604 T^{4} - 120 p T^{5} + 65 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 8 T - 19 T^{2} - 88 T^{3} + 1672 T^{4} - 88 p T^{5} - 19 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1264 T^{3} + 11806 T^{4} - 1264 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 45 T^{2} - 300 T^{3} - 5524 T^{4} - 300 p T^{5} + 45 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 30 T + 261 T^{2} + 702 T^{3} - 21664 T^{4} + 702 p T^{5} + 261 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 16 T + 65 T^{2} - 960 T^{3} - 14284 T^{4} - 960 p T^{5} + 65 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 23814 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 - 155 T^{2} + 17784 T^{4} - 155 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16 T + 185 T^{2} - 784 T^{3} + 4900 T^{4} - 784 p T^{5} + 185 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 13094 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.61485953140342362977510081610, −7.60861180980191805151075174702, −7.60151306850621739276078406874, −7.22627305520648180321460997345, −7.10188257370848706768414269321, −6.17218802157949538224783852568, −6.12887263880808359540798833493, −5.80376915092587888653365147576, −5.74145650076230080048615771115, −5.33866764512297021622194394154, −5.20466998171835004367448205588, −5.19796074638179476577494345650, −5.19019176032247911889527181993, −4.56378366603511762770671315633, −4.45559449432927182013915747684, −3.73008795082027087734488673489, −3.56018545535216238145527784744, −3.34440003674585134994328785119, −3.13318201204148951053041642960, −2.40543256818427037116487358503, −2.24284944436441471978316693036, −1.91409019923414559038292049707, −1.59734614208613301947217656165, −1.16025467023798581145218401595, −0.896052327697105328122559191309,
0.896052327697105328122559191309, 1.16025467023798581145218401595, 1.59734614208613301947217656165, 1.91409019923414559038292049707, 2.24284944436441471978316693036, 2.40543256818427037116487358503, 3.13318201204148951053041642960, 3.34440003674585134994328785119, 3.56018545535216238145527784744, 3.73008795082027087734488673489, 4.45559449432927182013915747684, 4.56378366603511762770671315633, 5.19019176032247911889527181993, 5.19796074638179476577494345650, 5.20466998171835004367448205588, 5.33866764512297021622194394154, 5.74145650076230080048615771115, 5.80376915092587888653365147576, 6.12887263880808359540798833493, 6.17218802157949538224783852568, 7.10188257370848706768414269321, 7.22627305520648180321460997345, 7.60151306850621739276078406874, 7.60861180980191805151075174702, 7.61485953140342362977510081610
