| L(s) = 1 | − 2·2-s + 4-s + 2·5-s + 2·8-s − 4·10-s + 2·11-s + 20·13-s − 4·16-s + 12·17-s + 6·19-s + 2·20-s − 4·22-s − 2·23-s + 4·25-s − 40·26-s − 4·29-s + 8·31-s + 2·32-s − 24·34-s − 2·37-s − 12·38-s + 4·40-s − 12·41-s − 16·43-s + 2·44-s + 4·46-s + 16·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 0.894·5-s + 0.707·8-s − 1.26·10-s + 0.603·11-s + 5.54·13-s − 16-s + 2.91·17-s + 1.37·19-s + 0.447·20-s − 0.852·22-s − 0.417·23-s + 4/5·25-s − 7.84·26-s − 0.742·29-s + 1.43·31-s + 0.353·32-s − 4.11·34-s − 0.328·37-s − 1.94·38-s + 0.632·40-s − 1.87·41-s − 2.43·43-s + 0.301·44-s + 0.589·46-s + 2.33·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.719396670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.719396670\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 2 T + 12 T^{3} - 29 T^{4} + 12 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T - 4 T^{2} - 12 T^{3} + 555 T^{4} - 12 p T^{5} - 4 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 2 T - 36 T^{2} - 12 T^{3} + 979 T^{4} - 12 p T^{5} - 36 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 2 T + 31 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 2 T - 64 T^{2} - 12 T^{3} + 3107 T^{4} - 12 p T^{5} - 64 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 - 16 T + 126 T^{2} - 576 T^{3} + 2659 T^{4} - 576 p T^{5} + 126 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 2 T - 96 T^{2} - 12 T^{3} + 6979 T^{4} - 12 p T^{5} - 96 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4 T - 99 T^{2} + 12 T^{3} + 8800 T^{4} + 12 p T^{5} - 99 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 18 T + 149 T^{2} + 954 T^{3} + 7140 T^{4} + 954 p T^{5} + 149 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T - 23 T^{2} + 376 T^{3} - 1208 T^{4} + 376 p T^{5} - 23 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 14 T + 184 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 6 T - 112 T^{2} + 12 T^{3} + 13947 T^{4} + 12 p T^{5} - 112 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 - 151 T^{2} + 16560 T^{4} - 151 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 8 T - 18 T^{2} - 768 T^{3} - 6893 T^{4} - 768 p T^{5} - 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 22 T + 287 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.95682920471338390867479368088, −6.70050936414774951729903961038, −6.20725411230432128902852773991, −6.08856887979021775552599421224, −5.95840829715130355860913176446, −5.87888907312900173231945641792, −5.74282989916066471022986272219, −5.51472189663691442290236474300, −5.17160470064471362199163279167, −4.85008144723029673143917371575, −4.55867703494728280491804111965, −4.39447154289855556606540193219, −3.79846787632667436640990619115, −3.76559883339007094640113604820, −3.70738455813615316773875832305, −3.32662839149176983205271139755, −3.05542150373545098341434143491, −3.01999055556669116457930318444, −2.61898972939481430675837603024, −1.64985462577061168119292547808, −1.53286420911197289120146696577, −1.46950536038448675635032126872, −1.36293796693553472806189891109, −1.06439894563318349657540342142, −0.53140231888941704425220584272,
0.53140231888941704425220584272, 1.06439894563318349657540342142, 1.36293796693553472806189891109, 1.46950536038448675635032126872, 1.53286420911197289120146696577, 1.64985462577061168119292547808, 2.61898972939481430675837603024, 3.01999055556669116457930318444, 3.05542150373545098341434143491, 3.32662839149176983205271139755, 3.70738455813615316773875832305, 3.76559883339007094640113604820, 3.79846787632667436640990619115, 4.39447154289855556606540193219, 4.55867703494728280491804111965, 4.85008144723029673143917371575, 5.17160470064471362199163279167, 5.51472189663691442290236474300, 5.74282989916066471022986272219, 5.87888907312900173231945641792, 5.95840829715130355860913176446, 6.08856887979021775552599421224, 6.20725411230432128902852773991, 6.70050936414774951729903961038, 6.95682920471338390867479368088
