| L(s) = 1 | + 232·4-s − 1.45e3·9-s − 9.22e3·11-s + 1.55e4·16-s + 4.78e4·19-s + 6.93e5·29-s − 2.58e5·31-s − 3.38e5·36-s − 6.81e5·41-s − 2.13e6·44-s + 3.24e6·49-s − 1.81e6·59-s − 9.56e5·61-s − 1.47e6·64-s − 5.21e6·71-s + 1.11e7·76-s − 1.24e7·79-s + 1.59e6·81-s − 3.27e7·89-s + 1.34e7·99-s − 4.55e7·101-s + 3.71e7·109-s + 1.60e8·116-s − 7.22e6·121-s − 5.99e7·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.81·4-s − 2/3·9-s − 2.08·11-s + 0.948·16-s + 1.60·19-s + 5.27·29-s − 1.55·31-s − 1.20·36-s − 1.54·41-s − 3.78·44-s + 3.93·49-s − 1.15·59-s − 0.539·61-s − 0.704·64-s − 1.72·71-s + 2.90·76-s − 2.84·79-s + 1/3·81-s − 4.91·89-s + 1.39·99-s − 4.40·101-s + 2.74·109-s + 9.56·116-s − 0.370·121-s − 2.82·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.3846859789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3846859789\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 29 p^{3} T^{2} + 2393 p^{4} T^{4} - 29 p^{17} T^{6} + p^{28} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 3241866 T^{2} + 3983690183803 T^{4} - 3241866 p^{14} T^{6} + p^{28} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 4612 T + 35518234 T^{2} + 4612 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 188603138 T^{2} + 15884000002783923 T^{4} - 188603138 p^{14} T^{6} + p^{28} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 231136540 T^{2} + 109118387602401158 T^{4} - 231136540 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 23934 T + 995928583 T^{2} - 23934 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 9685702964 T^{2} + 45905401251825777798 T^{4} - 9685702964 p^{14} T^{6} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 346508 T + 64488369118 T^{2} - 346508 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 129178 T + 29398278647 T^{2} + 129178 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 34059096724 T^{2} + \)\(17\!\cdots\!38\)\( T^{4} + 34059096724 p^{14} T^{6} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 340928 T + 203171618242 T^{2} + 340928 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 475958888570 T^{2} + \)\(19\!\cdots\!23\)\( T^{4} - 475958888570 p^{14} T^{6} + p^{28} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1863978803380 T^{2} + \)\(13\!\cdots\!38\)\( T^{4} - 1863978803380 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1059553636084 T^{2} + \)\(21\!\cdots\!78\)\( T^{4} - 1059553636084 p^{14} T^{6} + p^{28} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 907844 T + 3501429103018 T^{2} + 907844 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 478410 T - 2138250815357 T^{2} + 478410 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 14049180666090 T^{2} + \)\(12\!\cdots\!83\)\( T^{4} - 14049180666090 p^{14} T^{6} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 2607296 T + 17905773132286 T^{2} + 2607296 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 35080618298204 T^{2} + \)\(54\!\cdots\!58\)\( T^{4} - 35080618298204 p^{14} T^{6} + p^{28} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 6227360 T + 47990625121118 T^{2} + 6227360 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 49505183352548 T^{2} + \)\(13\!\cdots\!38\)\( T^{4} - 49505183352548 p^{14} T^{6} + p^{28} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 16356096 T + 133666580484178 T^{2} + 16356096 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 105137761003970 T^{2} + \)\(57\!\cdots\!63\)\( T^{4} - 105137761003970 p^{14} T^{6} + p^{28} T^{8} \) |
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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
−9.204269018106329150062148569955, −8.926492247153388406286236946505, −8.435036207006088340082379679683, −8.391775953328907320910763102097, −8.103363065944355400649924285767, −7.61724819395374531984633734425, −7.26250645232416088571229871624, −7.03510198594618798712833283773, −6.98822565574709188900298127304, −6.43332218056475699955620300391, −6.11024958504832026711128529619, −5.62074543651933436627331023046, −5.54230001984916523792012666510, −5.09260288322516408697862779374, −4.77447491652459104401299138416, −4.24925815523485444179559158555, −3.84070058027887085152140685242, −2.90955508959112175298462550149, −2.75597472497793872879218896709, −2.72147096715287104997604370064, −2.64448475569834236789546903373, −1.59894080764473113331254024960, −1.38977928232019042114277447432, −0.876746544664156877195897050110, −0.085392835771727571337748015367,
0.085392835771727571337748015367, 0.876746544664156877195897050110, 1.38977928232019042114277447432, 1.59894080764473113331254024960, 2.64448475569834236789546903373, 2.72147096715287104997604370064, 2.75597472497793872879218896709, 2.90955508959112175298462550149, 3.84070058027887085152140685242, 4.24925815523485444179559158555, 4.77447491652459104401299138416, 5.09260288322516408697862779374, 5.54230001984916523792012666510, 5.62074543651933436627331023046, 6.11024958504832026711128529619, 6.43332218056475699955620300391, 6.98822565574709188900298127304, 7.03510198594618798712833283773, 7.26250645232416088571229871624, 7.61724819395374531984633734425, 8.103363065944355400649924285767, 8.391775953328907320910763102097, 8.435036207006088340082379679683, 8.926492247153388406286236946505, 9.204269018106329150062148569955
