| L(s) = 1 | + 12·7-s − 4·13-s − 16·25-s + 24·31-s − 48·37-s − 12·43-s + 52·49-s − 16·61-s + 84·67-s − 96·73-s + 108·79-s − 48·91-s − 16·97-s − 24·109-s + 16·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 34·169-s + 173-s − 192·175-s + ⋯ |
| L(s) = 1 | + 4.53·7-s − 1.10·13-s − 3.19·25-s + 4.31·31-s − 7.89·37-s − 1.82·43-s + 52/7·49-s − 2.04·61-s + 10.2·67-s − 11.2·73-s + 12.1·79-s − 5.03·91-s − 1.62·97-s − 2.29·109-s + 1.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.61·169-s + 0.0760·173-s − 14.5·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.26959396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.26959396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 16 T^{2} + 29 p T^{4} + 976 T^{6} + 5296 T^{8} + 976 p^{2} T^{10} + 29 p^{5} T^{12} + 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 6 T + 4 p T^{2} - 96 T^{3} + 291 T^{4} - 96 p T^{5} + 4 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 16 T^{2} + 142 T^{4} + 2048 T^{6} - 32621 T^{8} + 2048 p^{2} T^{10} + 142 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 + 2 T - 11 T^{2} - 22 T^{3} + 4 T^{4} - 22 p T^{5} - 11 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 56 T^{2} + 1335 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 52 T^{2} + 1290 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 44 T^{2} + 1407 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 + 40 T^{2} - 407 T^{4} + 13000 T^{6} + 1975168 T^{8} + 13000 p^{2} T^{10} - 407 p^{4} T^{12} + 40 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 12 T + 106 T^{2} - 696 T^{3} + 3891 T^{4} - 696 p T^{5} + 106 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( 1 + 136 T^{2} + 10702 T^{4} + 602752 T^{6} + 26828899 T^{8} + 602752 p^{2} T^{10} + 10702 p^{4} T^{12} + 136 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 6 T + 52 T^{2} + 240 T^{3} + 267 T^{4} + 240 p T^{5} + 52 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 76 T^{2} + 1114 T^{4} - 18544 T^{6} + 4095379 T^{8} - 18544 p^{2} T^{10} + 1114 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 100 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( 1 + 68 T^{2} - 2294 T^{4} - 2992 T^{6} + 19645219 T^{8} - 2992 p^{2} T^{10} - 2294 p^{4} T^{12} + 68 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 8 T - 47 T^{2} - 88 T^{3} + 4696 T^{4} - 88 p T^{5} - 47 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 42 T + 868 T^{2} - 11760 T^{3} + 113307 T^{4} - 11760 p T^{5} + 868 p^{2} T^{6} - 42 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 152 T^{2} + 14130 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 24 T + 287 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 54 T + 1372 T^{2} - 21600 T^{3} + 230547 T^{4} - 21600 p T^{5} + 1372 p^{2} T^{6} - 54 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 220 T^{2} + 25594 T^{4} - 1986160 T^{6} + 139227715 T^{8} - 1986160 p^{2} T^{10} + 25594 p^{4} T^{12} - 220 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 128 T^{2} + 8031 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 8 T - 38 T^{2} - 736 T^{3} - 5213 T^{4} - 736 p T^{5} - 38 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.74129090185368473242214746724, −3.50251218859971900401200167251, −3.45329595377975553460451565254, −3.43556494889887153565472478914, −3.39633197680703095680971124766, −2.95335173106591659480895993398, −2.89436167728317346469383199362, −2.89386608657412599745947270371, −2.87124038128430749271498230477, −2.51739347473293342513889045653, −2.27781426978601865924831318885, −2.14312221560437478468116695043, −2.09288967472782566766320457578, −1.97095696990355508798869287563, −1.85939025211435157888091325794, −1.79205716243703924008495586767, −1.74434240069690911075688988717, −1.62067294608796109163379061787, −1.46011737973039548273251060221, −1.31645238095381842386324523367, −1.09417470922281486472975426559, −0.61099551946709706528241819940, −0.60314371251229060815810438833, −0.58784539513444943752777020085, −0.20775149752520766892665231550,
0.20775149752520766892665231550, 0.58784539513444943752777020085, 0.60314371251229060815810438833, 0.61099551946709706528241819940, 1.09417470922281486472975426559, 1.31645238095381842386324523367, 1.46011737973039548273251060221, 1.62067294608796109163379061787, 1.74434240069690911075688988717, 1.79205716243703924008495586767, 1.85939025211435157888091325794, 1.97095696990355508798869287563, 2.09288967472782566766320457578, 2.14312221560437478468116695043, 2.27781426978601865924831318885, 2.51739347473293342513889045653, 2.87124038128430749271498230477, 2.89386608657412599745947270371, 2.89436167728317346469383199362, 2.95335173106591659480895993398, 3.39633197680703095680971124766, 3.43556494889887153565472478914, 3.45329595377975553460451565254, 3.50251218859971900401200167251, 3.74129090185368473242214746724
Plot not available for L-functions of degree greater than 10.
