| L(s) = 1 | + 2·4-s − 6·5-s + 4·9-s − 4·13-s + 4·16-s + 6·19-s − 12·20-s + 6·23-s + 17·25-s + 12·29-s + 2·31-s + 8·36-s + 4·37-s + 24·41-s − 20·43-s − 24·45-s − 6·47-s − 8·52-s + 6·53-s − 12·59-s − 12·61-s + 16·64-s + 24·65-s + 12·67-s − 24·71-s + 10·73-s + 12·76-s + ⋯ |
| L(s) = 1 | + 4-s − 2.68·5-s + 4/3·9-s − 1.10·13-s + 16-s + 1.37·19-s − 2.68·20-s + 1.25·23-s + 17/5·25-s + 2.22·29-s + 0.359·31-s + 4/3·36-s + 0.657·37-s + 3.74·41-s − 3.04·43-s − 3.57·45-s − 0.875·47-s − 1.10·52-s + 0.824·53-s − 1.56·59-s − 1.53·61-s + 2·64-s + 2.97·65-s + 1.46·67-s − 2.84·71-s + 1.17·73-s + 1.37·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.591495667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.591495667\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 3 | $C_2^3$ | \( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - 4 T^{2} - 105 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 32 T^{2} + 735 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 7 T^{2} + 54 T^{3} - 204 T^{4} + 54 p T^{5} + 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T - 11 T^{2} - 6 T^{3} + 852 T^{4} - 6 p T^{5} - 11 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2 T - 41 T^{2} + 34 T^{3} + 940 T^{4} + 34 p T^{5} - 41 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T - 44 T^{2} + 56 T^{3} + 1639 T^{4} + 56 p T^{5} - 44 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 + 6 T - 65 T^{2} + 42 T^{3} + 6300 T^{4} + 42 p T^{5} - 65 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6 T - 71 T^{2} - 6 T^{3} + 6732 T^{4} - 6 p T^{5} - 71 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 22 T^{2} + 48 T^{3} + 2907 T^{4} + 48 p T^{5} + 22 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 46 T^{2} + 432 T^{3} - 4533 T^{4} + 432 p T^{5} + 46 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 128 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 10 T - 53 T^{2} - 70 T^{3} + 10780 T^{4} - 70 p T^{5} - 53 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 14 T + 61 T^{2} - 322 T^{3} - 3500 T^{4} - 322 p T^{5} + 61 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 18 T + 229 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T - 149 T^{2} + 42 T^{3} + 23100 T^{4} + 42 p T^{5} - 149 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 2 T + 33 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.55185644320047912807710415014, −7.52155129495034149648520088515, −7.30138030968960101635145609711, −6.93834892555442160158262301177, −6.83678929496537229541908280635, −6.52703594860114808257873580176, −6.31221564393132257067580687216, −6.20273345457375572043241372900, −5.65125628102944348544439401092, −5.28555950376612400791514332397, −5.14020520497608031039897388308, −4.85699648190543861347734672917, −4.51526380296329360908899372453, −4.43785922933592142099961430160, −4.21938404747790318117721286375, −3.79209419023328486459927886650, −3.57050807769704823976783886776, −3.25519199638047478425067199214, −2.94196093513247449332513168048, −2.70969450347938496056195857568, −2.57718109392760885401424828131, −1.75582133398507214641648096402, −1.48219492623847272883945898622, −0.843035439869330365332228107079, −0.63924813676598929352401460753,
0.63924813676598929352401460753, 0.843035439869330365332228107079, 1.48219492623847272883945898622, 1.75582133398507214641648096402, 2.57718109392760885401424828131, 2.70969450347938496056195857568, 2.94196093513247449332513168048, 3.25519199638047478425067199214, 3.57050807769704823976783886776, 3.79209419023328486459927886650, 4.21938404747790318117721286375, 4.43785922933592142099961430160, 4.51526380296329360908899372453, 4.85699648190543861347734672917, 5.14020520497608031039897388308, 5.28555950376612400791514332397, 5.65125628102944348544439401092, 6.20273345457375572043241372900, 6.31221564393132257067580687216, 6.52703594860114808257873580176, 6.83678929496537229541908280635, 6.93834892555442160158262301177, 7.30138030968960101635145609711, 7.52155129495034149648520088515, 7.55185644320047912807710415014
