| L(s) = 1 | − 356·2-s + 1.31e4·3-s + 1.07e5·4-s + 7.81e5·5-s − 4.67e6·6-s − 2.07e7·7-s − 7.84e7·8-s + 1.29e8·9-s − 2.78e8·10-s − 1.13e9·11-s + 1.41e9·12-s − 4.46e8·13-s + 7.38e9·14-s + 1.02e10·15-s + 2.53e10·16-s + 2.06e10·17-s − 4.59e10·18-s − 1.94e11·19-s + 8.43e10·20-s − 2.72e11·21-s + 4.02e11·22-s + 2.81e11·23-s − 1.02e12·24-s + 4.57e11·25-s + 1.59e11·26-s + 1.12e12·27-s − 2.24e12·28-s + ⋯ |
| L(s) = 1 | − 0.983·2-s + 1.15·3-s + 0.823·4-s + 0.894·5-s − 1.13·6-s − 1.36·7-s − 1.65·8-s + 9-s − 0.879·10-s − 1.59·11-s + 0.951·12-s − 0.151·13-s + 1.33·14-s + 1.03·15-s + 1.47·16-s + 0.718·17-s − 0.983·18-s − 2.62·19-s + 0.736·20-s − 1.57·21-s + 1.56·22-s + 0.750·23-s − 1.90·24-s + 3/5·25-s + 0.149·26-s + 0.769·27-s − 1.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{8} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{8} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 89 p^{2} T + 293 p^{6} T^{2} + 89 p^{19} T^{3} + p^{34} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2964936 p T + 10482058769006 p^{2} T^{2} + 2964936 p^{18} T^{3} + p^{34} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 1131629912 T + 1154204164608501734 T^{2} + 1131629912 p^{17} T^{3} + p^{34} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 446672524 T + 624610944442888038 p T^{2} + 446672524 p^{17} T^{3} + p^{34} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 20662021036 T + \)\(17\!\cdots\!78\)\( T^{2} - 20662021036 p^{17} T^{3} + p^{34} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 194376127216 T + \)\(19\!\cdots\!58\)\( T^{2} + 194376127216 p^{17} T^{3} + p^{34} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 281815042584 T + \)\(30\!\cdots\!46\)\( T^{2} - 281815042584 p^{17} T^{3} + p^{34} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4737062625892 T + \)\(13\!\cdots\!18\)\( T^{2} + 4737062625892 p^{17} T^{3} + p^{34} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2256407074872 T + \)\(39\!\cdots\!22\)\( T^{2} - 2256407074872 p^{17} T^{3} + p^{34} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 33686114979452 T + \)\(79\!\cdots\!14\)\( T^{2} + 33686114979452 p^{17} T^{3} + p^{34} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 13051545859228 T + \)\(52\!\cdots\!42\)\( T^{2} + 13051545859228 p^{17} T^{3} + p^{34} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 33149929851400 T + \)\(10\!\cdots\!90\)\( T^{2} + 33149929851400 p^{17} T^{3} + p^{34} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 84299585768440 T + \)\(44\!\cdots\!10\)\( T^{2} - 84299585768440 p^{17} T^{3} + p^{34} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 155836696419964 T - \)\(10\!\cdots\!94\)\( T^{2} - 155836696419964 p^{17} T^{3} + p^{34} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 963392423116456 T + \)\(25\!\cdots\!18\)\( T^{2} - 963392423116456 p^{17} T^{3} + p^{34} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2858793214972660 T + \)\(65\!\cdots\!18\)\( T^{2} + 2858793214972660 p^{17} T^{3} + p^{34} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6027859197776136 T + \)\(28\!\cdots\!78\)\( T^{2} - 6027859197776136 p^{17} T^{3} + p^{34} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2206536655060304 T - \)\(67\!\cdots\!14\)\( T^{2} - 2206536655060304 p^{17} T^{3} + p^{34} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 18626002136017276 T + \)\(24\!\cdots\!82\)\( p T^{2} + 18626002136017276 p^{17} T^{3} + p^{34} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4997891097934440 T + \)\(27\!\cdots\!18\)\( T^{2} - 4997891097934440 p^{17} T^{3} + p^{34} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 23351553482511288 T + \)\(54\!\cdots\!98\)\( T^{2} + 23351553482511288 p^{17} T^{3} + p^{34} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 60857257652216796 T + \)\(33\!\cdots\!78\)\( T^{2} + 60857257652216796 p^{17} T^{3} + p^{34} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 42869862659782756 T + \)\(14\!\cdots\!58\)\( T^{2} - 42869862659782756 p^{17} T^{3} + p^{34} T^{4} \) |
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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
−15.08974885145552067027826954698, −14.32572744533670611798593419045, −13.17906543526777722599927347812, −12.96179831693279076617456057667, −12.40162042529557743808462913412, −10.97650862413778942238206170504, −10.06848893939569723753244720963, −10.01220666376025558806813991229, −8.924554600898746256317057745820, −8.745350747949046446918439387358, −7.76089882839977728686029426365, −6.82218911287130375549995724698, −6.20556377686049608903008253241, −5.29767319805013976579092034304, −3.69708035424296950539879664778, −2.84017545143651075186639676318, −2.47722345968304888298274321211, −1.61711192957939357791759863108, 0, 0,
1.61711192957939357791759863108, 2.47722345968304888298274321211, 2.84017545143651075186639676318, 3.69708035424296950539879664778, 5.29767319805013976579092034304, 6.20556377686049608903008253241, 6.82218911287130375549995724698, 7.76089882839977728686029426365, 8.745350747949046446918439387358, 8.924554600898746256317057745820, 10.01220666376025558806813991229, 10.06848893939569723753244720963, 10.97650862413778942238206170504, 12.40162042529557743808462913412, 12.96179831693279076617456057667, 13.17906543526777722599927347812, 14.32572744533670611798593419045, 15.08974885145552067027826954698
