| L(s) = 1 | + 1.09e4·3-s + 7.81e5·5-s − 1.00e7·7-s − 8.76e7·9-s − 9.56e8·11-s − 1.23e9·13-s + 8.57e9·15-s − 2.72e10·17-s − 3.55e10·19-s − 1.10e11·21-s + 3.14e11·23-s + 4.57e11·25-s − 1.83e12·27-s + 4.91e11·29-s − 3.84e12·31-s − 1.05e13·33-s − 7.84e12·35-s − 2.83e13·37-s − 1.35e13·39-s − 7.18e13·41-s − 8.44e13·43-s − 6.84e13·45-s − 2.68e14·47-s − 3.62e14·49-s − 2.99e14·51-s − 4.07e14·53-s − 7.47e14·55-s + ⋯ |
| L(s) = 1 | + 0.966·3-s + 0.894·5-s − 0.658·7-s − 0.678·9-s − 1.34·11-s − 0.418·13-s + 0.864·15-s − 0.948·17-s − 0.479·19-s − 0.636·21-s + 0.838·23-s + 3/5·25-s − 1.24·27-s + 0.182·29-s − 0.808·31-s − 1.30·33-s − 0.589·35-s − 1.32·37-s − 0.404·39-s − 1.40·41-s − 1.10·43-s − 0.607·45-s − 1.64·47-s − 1.55·49-s − 0.916·51-s − 0.899·53-s − 1.20·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{8} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 1220 p^{2} T + 2570662 p^{4} T^{2} - 1220 p^{19} T^{3} + p^{34} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 1434980 p T + 1351062733890 p^{3} T^{2} + 1434980 p^{18} T^{3} + p^{34} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 956727120 T + 607481087904924742 T^{2} + 956727120 p^{17} T^{3} + p^{34} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 1231480820 T + 1193760722242242294 p T^{2} + 1231480820 p^{17} T^{3} + p^{34} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 27288798060 T + \)\(40\!\cdots\!58\)\( T^{2} + 27288798060 p^{17} T^{3} + p^{34} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 35527896968 T + \)\(11\!\cdots\!34\)\( T^{2} + 35527896968 p^{17} T^{3} + p^{34} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 314831989980 T + \)\(64\!\cdots\!70\)\( T^{2} - 314831989980 p^{17} T^{3} + p^{34} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 491350241052 T + \)\(52\!\cdots\!94\)\( T^{2} - 491350241052 p^{17} T^{3} + p^{34} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3840010967528 T + \)\(48\!\cdots\!18\)\( T^{2} + 3840010967528 p^{17} T^{3} + p^{34} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 28379654425940 T + \)\(10\!\cdots\!10\)\( T^{2} + 28379654425940 p^{17} T^{3} + p^{34} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 71857168552668 T + \)\(64\!\cdots\!18\)\( T^{2} + 71857168552668 p^{17} T^{3} + p^{34} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 84474258771260 T + \)\(13\!\cdots\!86\)\( T^{2} + 84474258771260 p^{17} T^{3} + p^{34} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 268222754609100 T + \)\(65\!\cdots\!18\)\( T^{2} + 268222754609100 p^{17} T^{3} + p^{34} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 407783381107140 T + \)\(42\!\cdots\!50\)\( T^{2} + 407783381107140 p^{17} T^{3} + p^{34} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1468770829559976 T + \)\(30\!\cdots\!82\)\( T^{2} + 1468770829559976 p^{17} T^{3} + p^{34} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 167375322642364 T + \)\(39\!\cdots\!66\)\( T^{2} - 167375322642364 p^{17} T^{3} + p^{34} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 5755588923766060 T + \)\(20\!\cdots\!70\)\( T^{2} - 5755588923766060 p^{17} T^{3} + p^{34} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2346325475416824 T - \)\(91\!\cdots\!74\)\( T^{2} + 2346325475416824 p^{17} T^{3} + p^{34} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12742204777546180 T + \)\(13\!\cdots\!82\)\( T^{2} - 12742204777546180 p^{17} T^{3} + p^{34} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 13823582395154896 T + \)\(33\!\cdots\!22\)\( T^{2} - 13823582395154896 p^{17} T^{3} + p^{34} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10669587622913700 T + \)\(85\!\cdots\!30\)\( T^{2} - 10669587622913700 p^{17} T^{3} + p^{34} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 26577073216629588 T + \)\(87\!\cdots\!94\)\( T^{2} - 26577073216629588 p^{17} T^{3} + p^{34} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 64801639405599020 T + \)\(12\!\cdots\!10\)\( T^{2} + 64801639405599020 p^{17} T^{3} + p^{34} T^{4} \) |
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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
−13.75000214785889953928102642993, −13.50566839163799753534975162156, −12.92221743583421887985575769322, −12.24962097744636024013938497515, −11.05999184712309551756252598652, −10.69397804016553702479541186421, −9.636037382444681390373801285272, −9.420951946369229954940494442285, −8.326441083619013406038161655388, −8.224099470825512187351694192783, −6.88106253009882143109064542170, −6.43661272546936414301766454959, −5.28043313866243136460256253459, −4.95098322516868625164621279063, −3.41004249911153316339023096293, −3.01279377607074751000554279053, −2.26286908905558293653623941911, −1.69154718610070591512577679734, 0, 0,
1.69154718610070591512577679734, 2.26286908905558293653623941911, 3.01279377607074751000554279053, 3.41004249911153316339023096293, 4.95098322516868625164621279063, 5.28043313866243136460256253459, 6.43661272546936414301766454959, 6.88106253009882143109064542170, 8.224099470825512187351694192783, 8.326441083619013406038161655388, 9.420951946369229954940494442285, 9.636037382444681390373801285272, 10.69397804016553702479541186421, 11.05999184712309551756252598652, 12.24962097744636024013938497515, 12.92221743583421887985575769322, 13.50566839163799753534975162156, 13.75000214785889953928102642993
