L(s) = 1 | + 2-s − 4·3-s − 4·6-s + 10·9-s + 8·11-s − 7·13-s + 16-s − 6·17-s + 10·18-s + 3·19-s + 8·22-s + 2·23-s − 7·26-s − 20·27-s + 16·29-s + 9·31-s + 3·32-s − 32·33-s − 6·34-s + 8·37-s + 3·38-s + 28·39-s + 4·41-s + 5·43-s + 2·46-s + 6·47-s − 4·48-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 2.30·3-s − 1.63·6-s + 10/3·9-s + 2.41·11-s − 1.94·13-s + 1/4·16-s − 1.45·17-s + 2.35·18-s + 0.688·19-s + 1.70·22-s + 0.417·23-s − 1.37·26-s − 3.84·27-s + 2.97·29-s + 1.61·31-s + 0.530·32-s − 5.57·33-s − 1.02·34-s + 1.31·37-s + 0.486·38-s + 4.48·39-s + 0.624·41-s + 0.762·43-s + 0.294·46-s + 0.875·47-s − 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.486680400\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.486680400\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$ | \( ( 1 + T )^{4} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 2 | $C_2 \wr S_4$ | \( 1 - T + T^{2} - T^{3} - p T^{5} + p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 8 T + 36 T^{2} - 84 T^{3} + 222 T^{4} - 84 p T^{5} + 36 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 7 T + 33 T^{2} + 94 T^{3} + 326 T^{4} + 94 p T^{5} + 33 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 6 T + 48 T^{2} + 198 T^{3} + 1094 T^{4} + 198 p T^{5} + 48 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 3 T + 39 T^{2} + 4 T^{3} + 564 T^{4} + 4 p T^{5} + 39 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 2 T + 24 T^{2} - 66 T^{3} + 1006 T^{4} - 66 p T^{5} + 24 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 9 T + 68 T^{2} - 261 T^{3} + 1526 T^{4} - 261 p T^{5} + 68 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 8 T + 86 T^{2} - 744 T^{3} + 4399 T^{4} - 744 p T^{5} + 86 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 4 T + 104 T^{2} - 256 T^{3} + 4966 T^{4} - 256 p T^{5} + 104 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 5 T + 120 T^{2} - 677 T^{3} + 6782 T^{4} - 677 p T^{5} + 120 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 6 T + 124 T^{2} - 346 T^{3} + 6382 T^{4} - 346 p T^{5} + 124 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 6 T + 136 T^{2} + 662 T^{3} + 9910 T^{4} + 662 p T^{5} + 136 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 10 T + 196 T^{2} - 1726 T^{3} + 16158 T^{4} - 1726 p T^{5} + 196 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - T + 251 T^{2} - 192 T^{3} + 24716 T^{4} - 192 p T^{5} + 251 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 22 T + 252 T^{2} - 1806 T^{3} + 12966 T^{4} - 1806 p T^{5} + 252 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 4 T + 170 T^{2} + 48 T^{3} + 12595 T^{4} + 48 p T^{5} + 170 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 8 T + 246 T^{2} - 1764 T^{3} + 26539 T^{4} - 1764 p T^{5} + 246 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 2 T + 220 T^{2} - 894 T^{3} + 22430 T^{4} - 894 p T^{5} + 220 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 10 T + 360 T^{2} + 2586 T^{3} + 48262 T^{4} + 2586 p T^{5} + 360 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 12 T + 330 T^{2} + 3296 T^{3} + 45123 T^{4} + 3296 p T^{5} + 330 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.05015808279899000557955730798, −6.04896198194130864069926149987, −5.51047493569894670753564173320, −5.43359279685886920294164906975, −5.21319285881945432896610320814, −4.92075844653154871535799541560, −4.78926866187320515935970520548, −4.65018318638189574374153111657, −4.61965412152586139591921273003, −4.21754204455376001262275304392, −4.20024372230889375065980994726, −3.95214555920067955812542912087, −3.91052004865100366151472266414, −3.28376190877692278832432796759, −3.19115287731094119644027302281, −2.93835039305876029940058074758, −2.42585238188988028512445655426, −2.41670714516299701658283727917, −2.38648685895019978029039672567, −1.69580234960839639722954779980, −1.42756271503770049357871000820, −1.21747382554731540025476394280, −0.894364077263901787210998350220, −0.54621061347035232346221105276, −0.51964981056468170057322385875,
0.51964981056468170057322385875, 0.54621061347035232346221105276, 0.894364077263901787210998350220, 1.21747382554731540025476394280, 1.42756271503770049357871000820, 1.69580234960839639722954779980, 2.38648685895019978029039672567, 2.41670714516299701658283727917, 2.42585238188988028512445655426, 2.93835039305876029940058074758, 3.19115287731094119644027302281, 3.28376190877692278832432796759, 3.91052004865100366151472266414, 3.95214555920067955812542912087, 4.20024372230889375065980994726, 4.21754204455376001262275304392, 4.61965412152586139591921273003, 4.65018318638189574374153111657, 4.78926866187320515935970520548, 4.92075844653154871535799541560, 5.21319285881945432896610320814, 5.43359279685886920294164906975, 5.51047493569894670753564173320, 6.04896198194130864069926149987, 6.05015808279899000557955730798