| L(s) = 1 | + 2·3-s + 4·7-s + 9-s + 2·13-s − 4·19-s + 8·21-s − 12·23-s + 4·25-s − 2·27-s + 28·31-s − 4·37-s + 4·39-s − 2·43-s + 12·47-s − 6·49-s − 8·57-s + 14·61-s + 4·63-s − 14·67-s − 24·69-s + 12·71-s + 14·73-s + 8·75-s + 10·79-s − 4·81-s + 48·83-s + 12·89-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s + 1/3·9-s + 0.554·13-s − 0.917·19-s + 1.74·21-s − 2.50·23-s + 4/5·25-s − 0.384·27-s + 5.02·31-s − 0.657·37-s + 0.640·39-s − 0.304·43-s + 1.75·47-s − 6/7·49-s − 1.05·57-s + 1.79·61-s + 0.503·63-s − 1.71·67-s − 2.88·69-s + 1.42·71-s + 1.63·73-s + 0.923·75-s + 1.12·79-s − 4/9·81-s + 5.26·83-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.902031993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.902031993\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 10 T^{2} - 189 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 68 T^{2} + 360 T^{3} + 1935 T^{4} + 360 p T^{5} + 68 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 34 T^{2} + 315 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 14 T + 105 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 2 T - 21 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 2 T - 77 T^{2} - 10 T^{3} + 4540 T^{4} - 10 p T^{5} - 77 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 38 T^{2} - 144 T^{3} + 2259 T^{4} - 144 p T^{5} + 38 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 100 T^{2} + 7191 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 64 T^{2} + 615 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 14 T + 49 T^{2} - 350 T^{3} + 5932 T^{4} - 350 p T^{5} + 49 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 14 T + 19 T^{2} + 602 T^{3} + 13708 T^{4} + 602 p T^{5} + 19 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 14 T + 25 T^{2} - 350 T^{3} + 9604 T^{4} - 350 p T^{5} + 25 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 10 T - 29 T^{2} + 290 T^{3} + 2500 T^{4} + 290 p T^{5} - 29 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 24 T + 286 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T - 64 T^{2} - 360 T^{3} + 22527 T^{4} - 360 p T^{5} - 64 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T + 70 T^{2} + 1600 T^{3} - 15581 T^{4} + 1600 p T^{5} + 70 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.27881255510789273247246691820, −6.93045729030021875951708113688, −6.64958854641558891239597174178, −6.61855358401815808870249128180, −6.37532541041620958400273667075, −6.24328130750888611111486786550, −6.00914755800263713063491048322, −5.41339275229641967926172574921, −5.30812710836862507055422689281, −5.30135228399889702556227531580, −4.61185770098675012012256275295, −4.61110894042898007873850410397, −4.59865998144501730143508344337, −3.99623223754965197268684183601, −3.87640158833013364517773406856, −3.80073618561518119352134233946, −3.15345430508960572516848221518, −3.13411261805055457493780790704, −2.56614463237603848284692972762, −2.45490227088569448173933109950, −2.11506353046408034267233005043, −1.96028686866628472435618425070, −1.45014052940401590642328373744, −0.978510660588083443643070857428, −0.63311125855415474990271379294,
0.63311125855415474990271379294, 0.978510660588083443643070857428, 1.45014052940401590642328373744, 1.96028686866628472435618425070, 2.11506353046408034267233005043, 2.45490227088569448173933109950, 2.56614463237603848284692972762, 3.13411261805055457493780790704, 3.15345430508960572516848221518, 3.80073618561518119352134233946, 3.87640158833013364517773406856, 3.99623223754965197268684183601, 4.59865998144501730143508344337, 4.61110894042898007873850410397, 4.61185770098675012012256275295, 5.30135228399889702556227531580, 5.30812710836862507055422689281, 5.41339275229641967926172574921, 6.00914755800263713063491048322, 6.24328130750888611111486786550, 6.37532541041620958400273667075, 6.61855358401815808870249128180, 6.64958854641558891239597174178, 6.93045729030021875951708113688, 7.27881255510789273247246691820
