| L(s) = 1 | − 330·5-s − 1.45e3·9-s + 1.26e4·11-s − 1.12e4·19-s − 4.63e4·25-s + 1.59e5·29-s + 3.05e5·31-s − 1.52e6·41-s + 4.81e5·45-s + 2.82e6·49-s − 4.18e6·55-s + 4.22e6·59-s + 1.57e6·61-s + 2.07e6·71-s + 8.10e6·79-s + 1.59e6·81-s − 2.32e7·89-s + 3.72e6·95-s − 1.84e7·99-s − 3.74e6·101-s − 2.88e5·109-s + 3.89e7·121-s + 4.07e7·125-s + ⋯ |
| L(s) = 1 | − 1.18·5-s − 2/3·9-s + 2.87·11-s − 0.377·19-s − 0.593·25-s + 1.21·29-s + 1.84·31-s − 3.45·41-s + 0.787·45-s + 3.43·49-s − 3.39·55-s + 2.67·59-s + 0.888·61-s + 0.687·71-s + 1.84·79-s + 1/3·81-s − 3.50·89-s + 0.445·95-s − 1.91·99-s − 0.361·101-s − 0.0213·109-s + 1.99·121-s + 1.86·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.947345249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.947345249\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 66 p T + 1242 p^{3} T^{2} + 66 p^{8} T^{3} + p^{14} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 2827304 T^{2} + 3331690026702 T^{4} - 2827304 p^{14} T^{6} + p^{28} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 6342 T + 40872558 T^{2} - 6342 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 215914768 T^{2} + 19245320323959534 T^{4} - 215914768 p^{14} T^{6} + p^{28} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 724051824 T^{2} + 258442067429732702 T^{4} - 724051824 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 5640 T + 1303903478 T^{2} + 5640 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 10658418300 T^{2} + 50149216274432940518 T^{4} - 10658418300 p^{14} T^{6} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 79758 T + 10563369034 T^{2} - 79758 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 152708 T + 2814513438 T^{2} - 152708 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 365391272720 T^{2} + \)\(51\!\cdots\!78\)\( T^{4} - 365391272720 p^{14} T^{6} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 761400 T + 521619097662 T^{2} + 761400 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 926857313900 T^{2} + \)\(35\!\cdots\!98\)\( T^{4} - 926857313900 p^{14} T^{6} + p^{28} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1641984198444 T^{2} + \)\(11\!\cdots\!22\)\( T^{4} - 1641984198444 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2736182196896 T^{2} + \)\(42\!\cdots\!42\)\( T^{4} - 2736182196896 p^{14} T^{6} + p^{28} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 2111178 T + 3567272493534 T^{2} - 2111178 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 787444 T + 6351476067726 T^{2} - 787444 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 2204173601956 T^{2} + \)\(74\!\cdots\!42\)\( T^{4} + 2204173601956 p^{14} T^{6} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 1036860 T + 7386973705582 T^{2} - 1036860 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 9396531758996 T^{2} + \)\(26\!\cdots\!22\)\( T^{4} - 9396531758996 p^{14} T^{6} + p^{28} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 4052436 T + 41254664174942 T^{2} - 4052436 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 63619047517260 T^{2} + \)\(20\!\cdots\!58\)\( T^{4} - 63619047517260 p^{14} T^{6} + p^{28} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 11647740 T + 115644497697558 T^{2} + 11647740 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 113234604744700 T^{2} + \)\(88\!\cdots\!38\)\( T^{4} + 113234604744700 p^{14} T^{6} + p^{28} T^{8} \) |
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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
−9.656957435652202587722755239801, −9.367151569384652933327257278171, −8.970538147288426875747131865110, −8.647157760112484211119124281279, −8.365670347955161772398748226187, −8.220645187691044722996828196467, −8.037015124229216311532446503795, −7.18601795763846449681825706990, −7.00901615127295757290555751801, −6.79780085851512336566648184061, −6.51491705263297147238981870108, −6.16614704890027753806871626545, −5.50281570749603002199260988362, −5.47194920631937205231352407823, −4.70292203031761695682844340478, −4.36924881996935474931969051849, −3.96199899823796946704257058854, −3.76277287429983481827629684383, −3.48253210132563495383149485864, −2.80137871096126676464539677710, −2.33596441919315233035371407163, −1.77666298309573489469911503085, −1.12650788757757722048260266922, −0.853919307543755543267963999541, −0.34832033413668753412487585162,
0.34832033413668753412487585162, 0.853919307543755543267963999541, 1.12650788757757722048260266922, 1.77666298309573489469911503085, 2.33596441919315233035371407163, 2.80137871096126676464539677710, 3.48253210132563495383149485864, 3.76277287429983481827629684383, 3.96199899823796946704257058854, 4.36924881996935474931969051849, 4.70292203031761695682844340478, 5.47194920631937205231352407823, 5.50281570749603002199260988362, 6.16614704890027753806871626545, 6.51491705263297147238981870108, 6.79780085851512336566648184061, 7.00901615127295757290555751801, 7.18601795763846449681825706990, 8.037015124229216311532446503795, 8.220645187691044722996828196467, 8.365670347955161772398748226187, 8.647157760112484211119124281279, 8.970538147288426875747131865110, 9.367151569384652933327257278171, 9.656957435652202587722755239801
