| L(s) = 1 | − 115·16-s − 1.80e3·43-s − 1.37e3·49-s − 3.39e3·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | − 1.79·16-s − 6.41·43-s − 4·49-s − 3.24·103-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + 0.000270·239-s + 0.000267·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + 115 T^{4} + p^{12} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 3742 T^{4} + p^{12} T^{8} \) |
| 7 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 11 | $D_4\times C_2$ | \( 1 - 1486370 T^{4} + p^{12} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 315317810 T^{4} + p^{12} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + 452 T + p^{3} T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 - 19480835090 T^{4} + p^{12} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 + 78746477470 T^{4} + p^{12} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 438410 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 234490873970 T^{4} + p^{12} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 811150 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 633473875010 T^{4} + p^{12} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 926581329650 T^{4} + p^{12} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
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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
−6.71455830302494320646054356260, −6.68613027365212190681432055115, −6.39890592584817335997313572899, −6.37291978815668365772005135821, −6.06178690426260958289221079158, −5.59841777465940521575435540804, −5.40733163364397188505797768215, −5.27584693661184407968128493907, −4.97316708748453382701676845352, −4.82528943994074521641636881511, −4.66267163451729961046086568386, −4.41636826877582856312733702961, −4.34302551539624240450587668861, −3.59149591061077614802528530241, −3.57470470352467855535306473403, −3.56493847822622501583319031591, −3.24681318391671947977094535206, −2.81889130521058540160829223361, −2.73129345536957542935029507039, −2.21194569838360631144232425907, −2.13409999330796236444640788122, −1.71403526455697741416538416985, −1.53515660561355164867993949043, −1.27193287679133555141862337288, −1.00429582563569572043564346386, 0, 0, 0, 0,
1.00429582563569572043564346386, 1.27193287679133555141862337288, 1.53515660561355164867993949043, 1.71403526455697741416538416985, 2.13409999330796236444640788122, 2.21194569838360631144232425907, 2.73129345536957542935029507039, 2.81889130521058540160829223361, 3.24681318391671947977094535206, 3.56493847822622501583319031591, 3.57470470352467855535306473403, 3.59149591061077614802528530241, 4.34302551539624240450587668861, 4.41636826877582856312733702961, 4.66267163451729961046086568386, 4.82528943994074521641636881511, 4.97316708748453382701676845352, 5.27584693661184407968128493907, 5.40733163364397188505797768215, 5.59841777465940521575435540804, 6.06178690426260958289221079158, 6.37291978815668365772005135821, 6.39890592584817335997313572899, 6.68613027365212190681432055115, 6.71455830302494320646054356260
