| L(s) = 1 | + 2·3-s − 2·4-s + 4·5-s + 6·7-s + 2·9-s − 4·12-s + 8·15-s + 3·16-s − 20·17-s − 8·20-s + 12·21-s + 6·27-s − 12·28-s + 24·35-s − 4·36-s − 4·41-s + 8·43-s + 8·45-s − 24·47-s + 6·48-s + 18·49-s − 40·51-s − 4·59-s − 16·60-s + 12·63-s − 4·64-s + 24·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s + 1.78·5-s + 2.26·7-s + 2/3·9-s − 1.15·12-s + 2.06·15-s + 3/4·16-s − 4.85·17-s − 1.78·20-s + 2.61·21-s + 1.15·27-s − 2.26·28-s + 4.05·35-s − 2/3·36-s − 0.624·41-s + 1.21·43-s + 1.19·45-s − 3.50·47-s + 0.866·48-s + 18/7·49-s − 5.60·51-s − 0.520·59-s − 2.06·60-s + 1.51·63-s − 1/2·64-s + 2.93·67-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
| \(L(1)\) |
\(\approx\) |
\(3.975472173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.975472173\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $D_{4}$ | \( ( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 2638 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_4\times C_2$ | \( 1 - 44 T^{2} + 886 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 4046 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 9038 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_4\times C_2$ | \( 1 - 76 T^{2} + 3766 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 16 T^{2} - 2354 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 6302 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 10 T + 178 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 89 | $D_{4}$ | \( ( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 340 T^{2} + 47398 T^{4} - 340 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.085715674385827048364887784384, −7.80384748991073404901300146209, −7.75038363231047214312353040547, −7.33520364373190921671750024876, −7.03846655063027790280926094730, −6.52340274352972771407586628821, −6.48434330685317520330089109800, −6.45924584654047377314713764410, −6.15756463831213322052217552619, −5.58969603402749060205599511720, −5.40355571960670569949362452388, −5.06036161139168043228720711218, −4.79691438553130064759370630154, −4.75220280849587052787640839500, −4.42047314677697039834379377798, −4.28314790339871214435564738376, −3.84863184689559725708360602731, −3.53387119496483252411691448233, −3.09039118620624704809392992406, −2.52226619264636754810946663350, −2.26513352989288105160917061919, −2.07172806355274297272936955619, −1.77206805510337242051100493331, −1.64435368882294884647650703222, −0.59837793069055945425910887008,
0.59837793069055945425910887008, 1.64435368882294884647650703222, 1.77206805510337242051100493331, 2.07172806355274297272936955619, 2.26513352989288105160917061919, 2.52226619264636754810946663350, 3.09039118620624704809392992406, 3.53387119496483252411691448233, 3.84863184689559725708360602731, 4.28314790339871214435564738376, 4.42047314677697039834379377798, 4.75220280849587052787640839500, 4.79691438553130064759370630154, 5.06036161139168043228720711218, 5.40355571960670569949362452388, 5.58969603402749060205599511720, 6.15756463831213322052217552619, 6.45924584654047377314713764410, 6.48434330685317520330089109800, 6.52340274352972771407586628821, 7.03846655063027790280926094730, 7.33520364373190921671750024876, 7.75038363231047214312353040547, 7.80384748991073404901300146209, 8.085715674385827048364887784384
