| L(s) = 1 | − 836·9-s + 7.29e3·11-s − 1.17e5·23-s − 1.87e5·25-s + 2.00e5·29-s + 4.30e5·37-s + 1.80e5·43-s + 2.09e6·53-s + 1.22e7·67-s + 1.06e7·71-s − 4.84e6·79-s + 6.21e6·81-s − 6.09e6·99-s − 3.35e7·107-s + 7.19e7·109-s + 4.31e7·113-s − 1.41e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.78e8·169-s + ⋯ |
| L(s) = 1 | − 0.382·9-s + 1.65·11-s − 2.01·23-s − 2.40·25-s + 1.52·29-s + 1.39·37-s + 0.346·43-s + 1.92·53-s + 4.95·67-s + 3.51·71-s − 1.10·79-s + 1.30·81-s − 0.631·99-s − 2.64·107-s + 5.32·109-s + 2.81·113-s − 0.726·121-s − 2.84·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(7.097159481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.097159481\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 836 T^{2} - 613322 p^{2} T^{4} + 836 p^{14} T^{6} + p^{28} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 37596 p T^{2} + 704301014 p^{2} T^{4} + 37596 p^{15} T^{6} + p^{28} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 3648 T + 27040758 T^{2} - 3648 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 178480556 T^{2} + 91261727628438 p^{2} T^{4} + 178480556 p^{14} T^{6} + p^{28} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 583939364 T^{2} + 401304026625255942 T^{4} + 583939364 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 261890044 T^{2} + 996315410356098726 T^{4} - 261890044 p^{14} T^{6} + p^{28} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 58920 T + 4243916494 T^{2} + 58920 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 100116 T + 33098852622 T^{2} - 100116 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 67824961244 T^{2} + \)\(22\!\cdots\!26\)\( T^{4} + 67824961244 p^{14} T^{6} + p^{28} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 215220 T + 2103598118 p T^{2} - 215220 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 61269104324 T^{2} + \)\(57\!\cdots\!66\)\( T^{4} + 61269104324 p^{14} T^{6} + p^{28} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 90320 T + 377428973814 T^{2} - 90320 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 828933408924 T^{2} + \)\(59\!\cdots\!82\)\( T^{4} + 828933408924 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 1045260 T + 1815280772574 T^{2} - 1045260 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 6070751087076 T^{2} + \)\(21\!\cdots\!66\)\( T^{4} + 6070751087076 p^{14} T^{6} + p^{28} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 1909232547884 T^{2} + \)\(19\!\cdots\!46\)\( T^{4} + 1909232547884 p^{14} T^{6} + p^{28} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 6102320 T + 21427566930246 T^{2} - 6102320 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 5305776 T + 23358636457326 T^{2} - 5305776 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 38817399588068 T^{2} + \)\(61\!\cdots\!74\)\( T^{4} + 38817399588068 p^{14} T^{6} + p^{28} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 2422096 T + 23621592577182 T^{2} + 2422096 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 76680260325188 T^{2} + \)\(26\!\cdots\!94\)\( T^{4} + 76680260325188 p^{14} T^{6} + p^{28} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 174170359946916 T^{2} + \)\(11\!\cdots\!46\)\( T^{4} + 174170359946916 p^{14} T^{6} + p^{28} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 124022836065764 T^{2} + \)\(10\!\cdots\!02\)\( T^{4} + 124022836065764 p^{14} T^{6} + p^{28} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.987347897488811387102179417936, −7.45706198588840037878187162448, −7.16287510710347989230690732475, −7.14225530107473719724447044280, −6.61932190824060201648077305858, −6.27267719001211712531528245398, −6.15266731621724019209367797773, −6.06231730402683645637877220655, −5.67989710613054600408633112942, −5.28792342004509464315831539655, −4.96953192124807551365851993796, −4.69409397474409752439679426854, −4.18597616266332587974506788399, −4.03307926078662361537070147093, −3.67322037775396802520427310153, −3.64562093916735774824776997908, −3.28901431838742372245257826127, −2.33908615703034950784756674411, −2.31476048096240339448748877336, −2.31376412147131358881579096165, −1.78357112734902401802742099782, −1.22600099730748843356733984561, −0.896532029685405410671454886676, −0.57651908996180597176475545151, −0.38703092985910797725974961583,
0.38703092985910797725974961583, 0.57651908996180597176475545151, 0.896532029685405410671454886676, 1.22600099730748843356733984561, 1.78357112734902401802742099782, 2.31376412147131358881579096165, 2.31476048096240339448748877336, 2.33908615703034950784756674411, 3.28901431838742372245257826127, 3.64562093916735774824776997908, 3.67322037775396802520427310153, 4.03307926078662361537070147093, 4.18597616266332587974506788399, 4.69409397474409752439679426854, 4.96953192124807551365851993796, 5.28792342004509464315831539655, 5.67989710613054600408633112942, 6.06231730402683645637877220655, 6.15266731621724019209367797773, 6.27267719001211712531528245398, 6.61932190824060201648077305858, 7.14225530107473719724447044280, 7.16287510710347989230690732475, 7.45706198588840037878187162448, 7.987347897488811387102179417936
