L(s) = 1 | + 64·4-s + 308·7-s + 2.00e3·9-s − 4.44e3·11-s + 3.07e3·16-s + 4.05e4·23-s + 1.09e4·25-s + 1.97e4·28-s − 1.82e4·29-s + 1.28e5·36-s − 2.31e4·37-s − 4.46e4·43-s − 2.84e5·44-s − 1.07e5·49-s + 2.48e5·53-s + 6.17e5·63-s + 1.31e5·64-s − 4.34e5·67-s − 4.51e5·71-s − 1.36e6·77-s + 2.09e6·79-s + 2.14e6·81-s + 2.59e6·92-s − 8.89e6·99-s + 6.97e5·100-s + 6.34e5·107-s + 5.82e6·109-s + ⋯ |
L(s) = 1 | + 4-s + 0.897·7-s + 2.74·9-s − 3.33·11-s + 3/4·16-s + 3.33·23-s + 0.697·25-s + 0.897·28-s − 0.748·29-s + 2.74·36-s − 0.457·37-s − 0.562·43-s − 3.33·44-s − 0.915·49-s + 1.66·53-s + 2.46·63-s + 1/2·64-s − 1.44·67-s − 1.26·71-s − 2.99·77-s + 4.24·79-s + 4.03·81-s + 3.33·92-s − 9.17·99-s + 0.697·100-s + 0.517·107-s + 4.49·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(3.837119394\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.837119394\) |
\(L(4)\) |
|
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 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 44 p T + 4134 p^{2} T^{2} - 44 p^{7} T^{3} + p^{12} T^{4} \) |
good | 3 | $C_2^2 \wr C_2$ | \( 1 - 668 p T^{2} + 208022 p^{2} T^{4} - 668 p^{13} T^{6} + p^{24} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 - 436 p^{2} T^{2} + 412902 p^{4} T^{4} - 436 p^{14} T^{6} + p^{24} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 2220 T + 4770614 T^{2} + 2220 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 3940180 T^{2} + 17707018928070 T^{4} - 3940180 p^{12} T^{6} + p^{24} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 56335492 T^{2} + 1681094907776646 T^{4} - 56335492 p^{12} T^{6} + p^{24} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 122207956 T^{2} + 7623942006639558 T^{4} - 122207956 p^{12} T^{6} + p^{24} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 20292 T + 396782822 T^{2} - 20292 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 9132 T - 168260074 T^{2} + 9132 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 1275081988 T^{2} + 1557093819506538246 T^{4} - 1275081988 p^{12} T^{6} + p^{24} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 11596 T + 3283526454 T^{2} + 11596 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 9929192068 T^{2} + 52045566850295970246 T^{4} - 9929192068 p^{12} T^{6} + p^{24} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 22348 T + 2121261174 T^{2} + 22348 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 22996383364 T^{2} + \)\(27\!\cdots\!18\)\( T^{4} - 22996383364 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 124308 T + 44342337206 T^{2} - 124308 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 11871974636 T^{2} + \)\(33\!\cdots\!86\)\( T^{4} + 11871974636 p^{12} T^{6} + p^{24} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 84307090516 T^{2} + \)\(41\!\cdots\!58\)\( T^{4} - 84307090516 p^{12} T^{6} + p^{24} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 217388 T + 192725823126 T^{2} + 217388 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 225804 T + 160756795334 T^{2} + 225804 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 377326906756 T^{2} + \)\(71\!\cdots\!98\)\( T^{4} - 377326906756 p^{12} T^{6} + p^{24} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 1046452 T + 754787202918 T^{2} - 1046452 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 828919590868 T^{2} + \)\(35\!\cdots\!70\)\( T^{4} - 828919590868 p^{12} T^{6} + p^{24} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 1729674521860 T^{2} + \)\(12\!\cdots\!30\)\( T^{4} - 1729674521860 p^{12} T^{6} + p^{24} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 3289276805380 T^{2} + \)\(40\!\cdots\!70\)\( T^{4} - 3289276805380 p^{12} T^{6} + p^{24} T^{8} \) |
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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
−13.56451171486002962272060594041, −13.09515784612772290387638888658, −12.85915816698137519740319576869, −12.61267282436283628360058488894, −12.30240115825826189790306003323, −11.47619625570003954078704736690, −11.04606202617613967104475756779, −10.94433243691035871113541021821, −10.34238995854355790171013150365, −10.24306775010029967900648671490, −9.954250041488454821248368793856, −9.023600513172322870039378606323, −8.743131243383779283035843669582, −7.85411874414407392759396515208, −7.56705771619632141047465505399, −7.39285511371144148459613847782, −6.97470933455247246630931941210, −6.33859245696523188220522060595, −5.21015436235711202081742289833, −5.04012306530305913138641290254, −4.69914401864769701379118414498, −3.44123114552857990266343436475, −2.65164762697882211504464742228, −1.86371346581523819161058772347, −0.961920847447946679456680070997,
0.961920847447946679456680070997, 1.86371346581523819161058772347, 2.65164762697882211504464742228, 3.44123114552857990266343436475, 4.69914401864769701379118414498, 5.04012306530305913138641290254, 5.21015436235711202081742289833, 6.33859245696523188220522060595, 6.97470933455247246630931941210, 7.39285511371144148459613847782, 7.56705771619632141047465505399, 7.85411874414407392759396515208, 8.743131243383779283035843669582, 9.023600513172322870039378606323, 9.954250041488454821248368793856, 10.24306775010029967900648671490, 10.34238995854355790171013150365, 10.94433243691035871113541021821, 11.04606202617613967104475756779, 11.47619625570003954078704736690, 12.30240115825826189790306003323, 12.61267282436283628360058488894, 12.85915816698137519740319576869, 13.09515784612772290387638888658, 13.56451171486002962272060594041