L(s) = 1 | − 2-s + 3-s − 5-s − 6-s − 7-s + 10-s + 11-s − 4·13-s + 14-s − 15-s + 4·17-s − 3·19-s − 21-s − 22-s + 18·23-s + 4·26-s − 14·29-s + 30-s + 9·31-s + 32-s + 33-s − 4·34-s + 35-s + 6·37-s + 3·38-s − 4·39-s + 8·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.316·10-s + 0.301·11-s − 1.10·13-s + 0.267·14-s − 0.258·15-s + 0.970·17-s − 0.688·19-s − 0.218·21-s − 0.213·22-s + 3.75·23-s + 0.784·26-s − 2.59·29-s + 0.182·30-s + 1.61·31-s + 0.176·32-s + 0.174·33-s − 0.685·34-s + 0.169·35-s + 0.986·37-s + 0.486·38-s − 0.640·39-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.763238888\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.763238888\) |
\(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 3 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 - T - 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
good | 5 | $C_2^2:C_4$ | \( 1 + T + T^{2} + 11 T^{3} + 36 T^{4} + 11 p T^{5} + p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 4 T + 3 T^{2} + 50 T^{3} + 341 T^{4} + 50 p T^{5} + 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 4 T - 11 T^{2} + 52 T^{3} + 69 T^{4} + 52 p T^{5} - 11 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 3 T + 35 T^{2} + 63 T^{3} + 784 T^{4} + 63 p T^{5} + 35 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 9 T + 65 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 + 14 T + 47 T^{2} - 208 T^{3} - 2275 T^{4} - 208 p T^{5} + 47 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 9 T + 5 T^{2} + 39 T^{3} + 484 T^{4} + 39 p T^{5} + 5 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 6 T - T^{2} - 192 T^{3} + 2449 T^{4} - 192 p T^{5} - p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} )( 1 + T - 69 T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 43 | $D_{4}$ | \( ( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 14 T + 29 T^{2} + 352 T^{3} - 2851 T^{4} + 352 p T^{5} + 29 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 18 T + 91 T^{2} - 114 T^{3} + 289 T^{4} - 114 p T^{5} + 91 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 10 T - 19 T^{2} + 10 p T^{3} - 3199 T^{4} + 10 p^{2} T^{5} - 19 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 - 12 T + 73 T^{2} - 24 T^{3} - 4895 T^{4} - 24 p T^{5} + 73 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 10 T + 87 T^{2} + 1130 T^{3} + 14489 T^{4} + 1130 p T^{5} + 87 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 14 T - 3 T^{2} - 608 T^{3} - 2275 T^{4} - 608 p T^{5} - 3 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 14 T + 193 T^{2} - 2140 T^{3} + 26421 T^{4} - 2140 p T^{5} + 193 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 13 T + 159 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 6 T - 21 T^{2} + 832 T^{3} + 14469 T^{4} + 832 p T^{5} - 21 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.107070677163660253101869672168, −7.74359973530997713774106734492, −7.50585039142048161943660190477, −7.15911413790482620125378650986, −7.09180175340071885164077669385, −7.08470808241160050925706982838, −6.58159352263424463045867180336, −6.55120882445338372889867370042, −5.82818174467965514728716438335, −5.67734535281926251695015590647, −5.48657669059509500897128805703, −5.45382355835451977281309901833, −4.82804651143583961419060707935, −4.65392226947357496225787681835, −4.25496581255278465012870330881, −4.18261287068864886038838632295, −3.63586497400685841627895988775, −3.47357608642884438622978628211, −3.11544074081659525400107140282, −2.69013473891280655609065381768, −2.44340959545908893695303794126, −2.32409322166496963259124028668, −1.52806388260460076721320882733, −0.854657942454831982931819724197, −0.71989202668836843885049436713,
0.71989202668836843885049436713, 0.854657942454831982931819724197, 1.52806388260460076721320882733, 2.32409322166496963259124028668, 2.44340959545908893695303794126, 2.69013473891280655609065381768, 3.11544074081659525400107140282, 3.47357608642884438622978628211, 3.63586497400685841627895988775, 4.18261287068864886038838632295, 4.25496581255278465012870330881, 4.65392226947357496225787681835, 4.82804651143583961419060707935, 5.45382355835451977281309901833, 5.48657669059509500897128805703, 5.67734535281926251695015590647, 5.82818174467965514728716438335, 6.55120882445338372889867370042, 6.58159352263424463045867180336, 7.08470808241160050925706982838, 7.09180175340071885164077669385, 7.15911413790482620125378650986, 7.50585039142048161943660190477, 7.74359973530997713774106734492, 8.107070677163660253101869672168