| L(s) = 1 | + 256·2-s + 4.91e4·4-s − 7.14e4·5-s − 8.96e5·7-s + 8.38e6·8-s − 1.82e7·10-s + 2.16e7·11-s − 1.51e8·13-s − 2.29e8·14-s + 1.34e9·16-s − 7.11e7·17-s + 1.41e9·19-s − 3.51e9·20-s + 5.53e9·22-s − 2.22e10·23-s − 2.54e10·25-s − 3.87e10·26-s − 4.40e10·28-s − 6.96e10·29-s − 1.87e11·31-s + 2.06e11·32-s − 1.82e10·34-s + 6.40e10·35-s − 7.21e11·37-s + 3.63e11·38-s − 5.99e11·40-s + 1.13e12·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.409·5-s − 0.411·7-s + 1.41·8-s − 0.578·10-s + 0.334·11-s − 0.668·13-s − 0.581·14-s + 5/4·16-s − 0.0420·17-s + 0.364·19-s − 0.613·20-s + 0.473·22-s − 1.36·23-s − 0.834·25-s − 0.946·26-s − 0.617·28-s − 0.749·29-s − 1.22·31-s + 1.06·32-s − 0.0594·34-s + 0.168·35-s − 1.24·37-s + 0.514·38-s − 0.578·40-s + 0.909·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+15/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{7} T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 71472 T + 6112016114 p T^{2} + 71472 p^{15} T^{3} + p^{30} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 128054 p T + 194036009127 p^{2} T^{2} + 128054 p^{16} T^{3} + p^{30} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 21628848 T + 717591876952562 p T^{2} - 21628848 p^{15} T^{3} + p^{30} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 11642174 p T + 296204149187259 p^{2} T^{2} + 11642174 p^{16} T^{3} + p^{30} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 71108496 T + 4854316123606278274 T^{2} + 71108496 p^{15} T^{3} + p^{30} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1418861578 T + 12463756324619774415 T^{2} - 1418861578 p^{15} T^{3} + p^{30} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 22216171152 T + \)\(46\!\cdots\!26\)\( T^{2} + 22216171152 p^{15} T^{3} + p^{30} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 69636203520 T + \)\(42\!\cdots\!42\)\( T^{2} + 69636203520 p^{15} T^{3} + p^{30} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 187701744632 T + \)\(47\!\cdots\!62\)\( T^{2} + 187701744632 p^{15} T^{3} + p^{30} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 721738522490 T + \)\(74\!\cdots\!95\)\( T^{2} + 721738522490 p^{15} T^{3} + p^{30} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 1134127875840 T + \)\(26\!\cdots\!98\)\( T^{2} - 1134127875840 p^{15} T^{3} + p^{30} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 735115869968 T + \)\(49\!\cdots\!06\)\( T^{2} + 735115869968 p^{15} T^{3} + p^{30} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1143668814288 T + \)\(23\!\cdots\!22\)\( T^{2} + 1143668814288 p^{15} T^{3} + p^{30} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7010634953376 T + \)\(15\!\cdots\!34\)\( T^{2} + 7010634953376 p^{15} T^{3} + p^{30} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4464695524944 T + \)\(72\!\cdots\!46\)\( T^{2} + 4464695524944 p^{15} T^{3} + p^{30} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 25346208518402 T + \)\(11\!\cdots\!47\)\( T^{2} + 25346208518402 p^{15} T^{3} + p^{30} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 36471044633486 T + \)\(63\!\cdots\!35\)\( T^{2} + 36471044633486 p^{15} T^{3} + p^{30} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 32664609209376 T + \)\(86\!\cdots\!02\)\( T^{2} + 32664609209376 p^{15} T^{3} + p^{30} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 57989177090770 T + \)\(13\!\cdots\!23\)\( T^{2} - 57989177090770 p^{15} T^{3} + p^{30} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 278861646180442 T + \)\(70\!\cdots\!39\)\( T^{2} - 278861646180442 p^{15} T^{3} + p^{30} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 493886855787936 T + \)\(18\!\cdots\!22\)\( T^{2} + 493886855787936 p^{15} T^{3} + p^{30} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1075161917501136 T + \)\(61\!\cdots\!98\)\( T^{2} + 1075161917501136 p^{15} T^{3} + p^{30} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 287158074477898 T + \)\(51\!\cdots\!51\)\( T^{2} - 287158074477898 p^{15} T^{3} + p^{30} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.96448414227724838418526662993, −11.63025870407325019964673943508, −11.00448065274807454033980208050, −10.38737330402729786041415381325, −9.622529173530973328667611113947, −9.234175120157917588498528929720, −7.918109623201612340629617498086, −7.87038742011005945753010414158, −6.87138946994295708105965932673, −6.56211857096733971932590502973, −5.57652646430564801534335880645, −5.42496374545260946275153892109, −4.34771901896551557345176061461, −4.05125827472453878583596693859, −3.31775944207364001978522456350, −2.81998595879653182618953971062, −1.82989081294112533973695313413, −1.53594664255397152006756341355, 0, 0,
1.53594664255397152006756341355, 1.82989081294112533973695313413, 2.81998595879653182618953971062, 3.31775944207364001978522456350, 4.05125827472453878583596693859, 4.34771901896551557345176061461, 5.42496374545260946275153892109, 5.57652646430564801534335880645, 6.56211857096733971932590502973, 6.87138946994295708105965932673, 7.87038742011005945753010414158, 7.918109623201612340629617498086, 9.234175120157917588498528929720, 9.622529173530973328667611113947, 10.38737330402729786041415381325, 11.00448065274807454033980208050, 11.63025870407325019964673943508, 11.96448414227724838418526662993
