| L(s) = 1 | + 18·4-s + 88·9-s − 208·11-s + 139·16-s − 32·29-s − 928·31-s + 1.58e3·36-s − 360·41-s − 3.74e3·44-s + 1.12e3·49-s + 960·59-s + 32·61-s + 324·64-s − 960·71-s − 912·79-s + 4.44e3·81-s − 536·89-s − 1.83e4·99-s + 3.14e3·101-s + 5.86e3·109-s − 576·116-s + 2.17e4·121-s − 1.67e4·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 9/4·4-s + 3.25·9-s − 5.70·11-s + 2.17·16-s − 0.204·29-s − 5.37·31-s + 22/3·36-s − 1.37·41-s − 12.8·44-s + 3.27·49-s + 2.11·59-s + 0.0671·61-s + 0.632·64-s − 1.60·71-s − 1.29·79-s + 6.09·81-s − 0.638·89-s − 18.5·99-s + 3.09·101-s + 5.15·109-s − 0.461·116-s + 16.3·121-s − 12.0·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.418086533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.418086533\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 9 p T^{2} + 185 T^{4} - 9 p^{7} T^{6} + p^{12} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3298 T^{4} - 88 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 1124 T^{2} + 541542 T^{4} - 1124 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 104 T + 5360 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 2868 T^{2} + 21712358 T^{4} + 2868 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 4592 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 968 T^{2} + 292876818 T^{4} - 968 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 16 T - 37558 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 464 T + 112392 T^{2} + 464 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 187940 T^{2} + 13919996118 T^{4} - 187940 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 180 T + 141886 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 251672 T^{2} + 28470270210 T^{4} - 251672 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 261828 T^{2} + 33667596230 T^{4} - 261828 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 148668 T^{2} + 3111063638 T^{4} - 148668 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 480 T + 447472 T^{2} - 480 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 16 T - 29910 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1153124 T^{2} + 513049854582 T^{4} - 1153124 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 480 T + 735976 T^{2} + 480 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 1009300 T^{2} + 482612715078 T^{4} - 1009300 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 456 T + 856406 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1995060 T^{2} + 1628060847638 T^{4} - 1995060 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 268 T + 1372598 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 3386876 T^{2} + 4533223381638 T^{4} - 3386876 p^{6} T^{6} + p^{12} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.72857752324124309264430767869, −7.61517072932110803098727114474, −7.55735464669137802168111044858, −7.21497002656106435400552658489, −7.03544844637951402148207449152, −6.96890887511855505825899217823, −6.77769722447203102869508013799, −6.02824751559700191592237673044, −5.69579559337930771250123415777, −5.63654576092001993346001585747, −5.37191671911010273291954992824, −5.30136933146310051259134136897, −4.72719399478571764031103646457, −4.57162292855796076269786541116, −4.22425669859562043449676352015, −3.66457056059367991234163244694, −3.45258569962173866298402615117, −3.01383603465115813456995915254, −2.77490412392478584231968311503, −2.25498292165912160894405140936, −1.94788788531348084050890706528, −1.89019228786998752282116869757, −1.82805925357178112798010876970, −0.60450528565830837476262984568, −0.43948612975675241840669240613,
0.43948612975675241840669240613, 0.60450528565830837476262984568, 1.82805925357178112798010876970, 1.89019228786998752282116869757, 1.94788788531348084050890706528, 2.25498292165912160894405140936, 2.77490412392478584231968311503, 3.01383603465115813456995915254, 3.45258569962173866298402615117, 3.66457056059367991234163244694, 4.22425669859562043449676352015, 4.57162292855796076269786541116, 4.72719399478571764031103646457, 5.30136933146310051259134136897, 5.37191671911010273291954992824, 5.63654576092001993346001585747, 5.69579559337930771250123415777, 6.02824751559700191592237673044, 6.77769722447203102869508013799, 6.96890887511855505825899217823, 7.03544844637951402148207449152, 7.21497002656106435400552658489, 7.55735464669137802168111044858, 7.61517072932110803098727114474, 7.72857752324124309264430767869
