| L(s) = 1 | + 483·4-s − 1.31e4·9-s − 4.30e4·11-s + 1.71e5·16-s + 1.91e5·19-s + 5.35e6·29-s + 2.15e7·31-s − 6.33e6·36-s + 5.21e7·41-s − 2.07e7·44-s + 7.42e6·49-s + 7.09e7·59-s + 6.82e8·61-s + 1.79e8·64-s + 4.20e8·71-s + 9.26e7·76-s + 4.95e7·79-s + 1.29e8·81-s + 8.55e8·89-s + 5.64e8·99-s + 2.71e7·101-s − 8.79e9·109-s + 2.58e9·116-s − 2.19e9·121-s + 1.04e10·124-s + ⋯ |
| L(s) = 1 | + 0.943·4-s − 2/3·9-s − 0.886·11-s + 0.655·16-s + 0.337·19-s + 1.40·29-s + 4.19·31-s − 0.628·36-s + 2.88·41-s − 0.835·44-s + 0.183·49-s + 0.762·59-s + 6.31·61-s + 1.34·64-s + 1.96·71-s + 0.318·76-s + 0.143·79-s + 1/3·81-s + 1.44·89-s + 0.590·99-s + 0.0260·101-s − 5.97·109-s + 1.32·116-s − 0.930·121-s + 3.95·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(7.230022995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.230022995\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 483 T^{2} + 15377 p^{2} T^{4} - 483 p^{18} T^{6} + p^{36} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 151516 p^{2} T^{2} - 894549623322 p^{4} T^{4} - 151516 p^{20} T^{6} + p^{36} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 21512 T + 1790961254 T^{2} + 21512 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 11151625772 T^{2} + \)\(24\!\cdots\!78\)\( T^{4} - 11151625772 p^{18} T^{6} + p^{36} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 400705004220 T^{2} + \)\(67\!\cdots\!18\)\( T^{4} - 400705004220 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 95896 T + 629883192438 T^{2} - 95896 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6185762667356 T^{2} + \)\(15\!\cdots\!18\)\( T^{4} - 6185762667356 p^{18} T^{6} + p^{36} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 2678212 T + 15397908029438 T^{2} - 2678212 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 10782432 T + 69294691361342 T^{2} - 10782432 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 235813135792844 T^{2} + \)\(37\!\cdots\!18\)\( T^{4} - 235813135792844 p^{18} T^{6} + p^{36} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 26060372 T + 693239183881142 T^{2} - 26060372 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1449556389870860 T^{2} + \)\(10\!\cdots\!98\)\( T^{4} - 1449556389870860 p^{18} T^{6} + p^{36} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 233226346198980 T^{2} + \)\(41\!\cdots\!78\)\( T^{4} + 233226346198980 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 13159896059574156 T^{2} + \)\(65\!\cdots\!18\)\( T^{4} - 13159896059574156 p^{18} T^{6} + p^{36} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 35494664 T + 7388006896329158 T^{2} - 35494664 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 341497340 T + 52053805546777278 T^{2} - 341497340 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 66154849722487660 T^{2} + \)\(25\!\cdots\!18\)\( T^{4} - 66154849722487660 p^{18} T^{6} + p^{36} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 210286064 T + 91549406631588686 T^{2} - 210286064 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 206940304125633116 T^{2} + \)\(17\!\cdots\!78\)\( T^{4} - 206940304125633116 p^{18} T^{6} + p^{36} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 24755040 T + 43090694479668638 T^{2} - 24755040 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 621372529743013292 T^{2} + \)\(16\!\cdots\!98\)\( T^{4} - 621372529743013292 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 427639116 T + 702028302670151638 T^{2} - 427639116 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1275322199616361340 T^{2} + \)\(12\!\cdots\!78\)\( T^{4} - 1275322199616361340 p^{18} T^{6} + p^{36} T^{8} \) |
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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
−8.788897106427554398606071864197, −8.269319783872599733219641871785, −8.100474032553013780017361395578, −8.015494610309467032967002268475, −7.895932646230383419699520380400, −7.19736457276668087501201360164, −6.78254154606258005069166673762, −6.63013155020538677973842808952, −6.49281417684530619765763004953, −6.08976089950273889620342779927, −5.52081291131769634854249212791, −5.38198236851038582819953532154, −5.06134848634657493680892311398, −4.69644514636486380669805167969, −4.15778731862293024547617088446, −3.69157841466815926839114600967, −3.68115899433781555919073294354, −2.59892370285273864769782292847, −2.57145220611146377447161179544, −2.54600413323381035291697880737, −2.37678603061590733578173276461, −1.20540842692248269925879300329, −0.988666208855612042954269633854, −0.900175602475609027889795867961, −0.34921553033541188286077896034,
0.34921553033541188286077896034, 0.900175602475609027889795867961, 0.988666208855612042954269633854, 1.20540842692248269925879300329, 2.37678603061590733578173276461, 2.54600413323381035291697880737, 2.57145220611146377447161179544, 2.59892370285273864769782292847, 3.68115899433781555919073294354, 3.69157841466815926839114600967, 4.15778731862293024547617088446, 4.69644514636486380669805167969, 5.06134848634657493680892311398, 5.38198236851038582819953532154, 5.52081291131769634854249212791, 6.08976089950273889620342779927, 6.49281417684530619765763004953, 6.63013155020538677973842808952, 6.78254154606258005069166673762, 7.19736457276668087501201360164, 7.895932646230383419699520380400, 8.015494610309467032967002268475, 8.100474032553013780017361395578, 8.269319783872599733219641871785, 8.788897106427554398606071864197
