| L(s) = 1 | − 8.04e5·9-s + 4.06e6·17-s − 2.07e8·25-s − 2.79e9·41-s − 4.58e9·49-s + 3.86e10·73-s + 3.07e11·81-s + 1.38e11·89-s − 1.01e11·97-s − 1.25e11·113-s − 1.79e12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 3.26e12·153-s + 157-s + 163-s + 167-s − 8.58e12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 4.53·9-s + 0.694·17-s − 4.25·25-s − 3.76·41-s − 2.32·49-s + 2.18·73-s + 9.78·81-s + 2.63·89-s − 1.19·97-s − 0.643·113-s − 6.30·121-s − 3.15·153-s − 4.79·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( ( 1 + 44668 p^{2} T^{2} + 121844326 p^{6} T^{4} + 44668 p^{24} T^{6} + p^{44} T^{8} )^{2} \) |
| 5 | \( ( 1 + 20778724 p T^{2} + 43226075612526 p^{3} T^{4} + 20778724 p^{23} T^{6} + p^{44} T^{8} )^{2} \) |
| 7 | \( ( 1 + 327807940 p T^{2} + 192177018862252266 p T^{4} + 327807940 p^{23} T^{6} + p^{44} T^{8} )^{2} \) |
| 11 | \( ( 1 + 899632022716 T^{2} + \)\(36\!\cdots\!10\)\( T^{4} + 899632022716 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 13 | \( ( 1 + 4293783272404 T^{2} + \)\(89\!\cdots\!58\)\( T^{4} + 4293783272404 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 17 | \( ( 1 - 1016268 T + 68151802908166 T^{2} - 1016268 p^{11} T^{3} + p^{22} T^{4} )^{4} \) |
| 19 | \( ( 1 + 394918064082076 T^{2} + \)\(65\!\cdots\!66\)\( T^{4} + 394918064082076 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 23 | \( ( 1 + 36709893265732 p T^{2} + \)\(11\!\cdots\!06\)\( T^{4} + 36709893265732 p^{23} T^{6} + p^{44} T^{8} )^{2} \) |
| 29 | \( ( 1 + 45883780928217236 T^{2} + \)\(82\!\cdots\!06\)\( T^{4} + 45883780928217236 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 31 | \( ( 1 + 4341089601754492 T^{2} + \)\(10\!\cdots\!82\)\( T^{4} + 4341089601754492 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 37 | \( ( 1 + 183804508815907828 T^{2} + \)\(49\!\cdots\!90\)\( T^{4} + 183804508815907828 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 41 | \( ( 1 + 698373012 T + 339100070273501782 T^{2} + 698373012 p^{11} T^{3} + p^{22} T^{4} )^{4} \) |
| 43 | \( ( 1 + 3319428097393245628 T^{2} + \)\(44\!\cdots\!50\)\( T^{4} + 3319428097393245628 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 47 | \( ( 1 + 2823743818210012220 T^{2} + \)\(14\!\cdots\!22\)\( T^{4} + 2823743818210012220 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 53 | \( ( 1 + 1711317589189151924 T^{2} + \)\(15\!\cdots\!38\)\( T^{4} + 1711317589189151924 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 59 | \( ( 1 + 73871288556722114428 T^{2} + \)\(29\!\cdots\!42\)\( T^{4} + 73871288556722114428 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 61 | \( ( 1 + 1898962485795286948 p T^{2} + \)\(63\!\cdots\!54\)\( T^{4} + 1898962485795286948 p^{23} T^{6} + p^{44} T^{8} )^{2} \) |
| 67 | \( ( 1 + \)\(30\!\cdots\!92\)\( T^{2} + \)\(53\!\cdots\!50\)\( T^{4} + \)\(30\!\cdots\!92\)\( p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 71 | \( ( 1 + \)\(47\!\cdots\!12\)\( T^{2} + \)\(16\!\cdots\!22\)\( T^{4} + \)\(47\!\cdots\!12\)\( p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 73 | \( ( 1 - 9662070436 T + \)\(63\!\cdots\!54\)\( T^{2} - 9662070436 p^{11} T^{3} + p^{22} T^{4} )^{4} \) |
| 79 | \( ( 1 + \)\(20\!\cdots\!16\)\( T^{2} + \)\(20\!\cdots\!46\)\( T^{4} + \)\(20\!\cdots\!16\)\( p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 83 | \( ( 1 + \)\(33\!\cdots\!08\)\( T^{2} + \)\(58\!\cdots\!10\)\( T^{4} + \)\(33\!\cdots\!08\)\( p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 89 | \( ( 1 - 34729039620 T + \)\(48\!\cdots\!62\)\( T^{2} - 34729039620 p^{11} T^{3} + p^{22} T^{4} )^{4} \) |
| 97 | \( ( 1 + 25343172500 T + \)\(86\!\cdots\!06\)\( T^{2} + 25343172500 p^{11} T^{3} + p^{22} T^{4} )^{4} \) |
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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
−4.12752070014524948637721031208, −3.74513112055978883162024319178, −3.55354676052625445891219650790, −3.52963813547394271403930913526, −3.47232872148836829107177404792, −3.46827456011030376546514218557, −3.35891810238211645886200452653, −3.30479961722704989217028875079, −3.26958195801718573071150338423, −2.64426327483114646770611691606, −2.56245137040586394441366523036, −2.53829005728859010748026616036, −2.45639096797983856890014101446, −2.32173126743261746169982632706, −2.27413910484661906734909386797, −2.11280952922799727759645126938, −1.98981904679326847522979816231, −1.72029826029608130840802802667, −1.48323487017928003683160436716, −1.33845825002005586526463352197, −1.23255772211946081996682597817, −1.15154256988131611190386566003, −1.05546052209206020841175045279, −0.806110306131805757101222725229, −0.69439969980969944508249958343, 0, 0, 0, 0, 0, 0, 0, 0,
0.69439969980969944508249958343, 0.806110306131805757101222725229, 1.05546052209206020841175045279, 1.15154256988131611190386566003, 1.23255772211946081996682597817, 1.33845825002005586526463352197, 1.48323487017928003683160436716, 1.72029826029608130840802802667, 1.98981904679326847522979816231, 2.11280952922799727759645126938, 2.27413910484661906734909386797, 2.32173126743261746169982632706, 2.45639096797983856890014101446, 2.53829005728859010748026616036, 2.56245137040586394441366523036, 2.64426327483114646770611691606, 3.26958195801718573071150338423, 3.30479961722704989217028875079, 3.35891810238211645886200452653, 3.46827456011030376546514218557, 3.47232872148836829107177404792, 3.52963813547394271403930913526, 3.55354676052625445891219650790, 3.74513112055978883162024319178, 4.12752070014524948637721031208
Plot not available for L-functions of degree greater than 10.
