| L(s) = 1 | − 1.31e4·3-s + 3.82e5·5-s − 2.44e7·7-s + 1.29e8·9-s + 9.87e8·11-s − 2.51e9·13-s − 5.02e9·15-s − 3.43e10·17-s − 8.00e10·19-s + 3.21e11·21-s − 2.97e11·23-s + 2.08e11·25-s − 1.12e12·27-s − 4.70e11·29-s − 3.40e12·31-s − 1.29e13·33-s − 9.36e12·35-s + 1.06e13·37-s + 3.30e13·39-s − 1.13e14·41-s − 6.16e13·43-s + 4.94e13·45-s + 2.79e14·47-s + 9.70e13·49-s + 4.50e14·51-s − 5.30e14·53-s + 3.78e14·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.438·5-s − 1.60·7-s + 9-s + 1.38·11-s − 0.856·13-s − 0.506·15-s − 1.19·17-s − 1.08·19-s + 1.85·21-s − 0.791·23-s + 0.273·25-s − 0.769·27-s − 0.174·29-s − 0.716·31-s − 1.60·33-s − 0.703·35-s + 0.498·37-s + 0.989·39-s − 2.21·41-s − 0.804·43-s + 0.438·45-s + 1.71·47-s + 0.417·49-s + 1.37·51-s − 1.17·53-s + 0.608·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.8197278801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8197278801\) |
| \(L(\frac{19}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p^{8} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 76572 p T - 496938298 p^{3} T^{2} - 76572 p^{18} T^{3} + p^{34} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 24471568 T + 10241319447726 p^{2} T^{2} + 24471568 p^{17} T^{3} + p^{34} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 89777592 p T + 10351710384952534 p^{2} T^{2} - 89777592 p^{18} T^{3} + p^{34} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2519398244 T + 1337435266843228758 p T^{2} + 2519398244 p^{17} T^{3} + p^{34} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 34313126364 T + \)\(18\!\cdots\!78\)\( T^{2} + 34313126364 p^{17} T^{3} + p^{34} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 80053542184 T + \)\(10\!\cdots\!58\)\( T^{2} + 80053542184 p^{17} T^{3} + p^{34} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 297228742704 T + \)\(59\!\cdots\!86\)\( T^{2} + 297228742704 p^{17} T^{3} + p^{34} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 470374069572 T + \)\(14\!\cdots\!38\)\( T^{2} + 470374069572 p^{17} T^{3} + p^{34} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3400754454592 T + \)\(45\!\cdots\!22\)\( T^{2} + 3400754454592 p^{17} T^{3} + p^{34} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 10652012180428 T + \)\(49\!\cdots\!14\)\( T^{2} - 10652012180428 p^{17} T^{3} + p^{34} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 113376799448748 T + \)\(78\!\cdots\!02\)\( T^{2} + 113376799448748 p^{17} T^{3} + p^{34} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 61637031489880 T + \)\(81\!\cdots\!90\)\( T^{2} + 61637031489880 p^{17} T^{3} + p^{34} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 279645641926560 T + \)\(72\!\cdots\!10\)\( T^{2} - 279645641926560 p^{17} T^{3} + p^{34} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 530964038611476 T + \)\(46\!\cdots\!46\)\( T^{2} + 530964038611476 p^{17} T^{3} + p^{34} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1727524231086456 T + \)\(23\!\cdots\!58\)\( T^{2} + 1727524231086456 p^{17} T^{3} + p^{34} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2784287656027900 T + \)\(64\!\cdots\!98\)\( T^{2} - 2784287656027900 p^{17} T^{3} + p^{34} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3329301676696184 T + \)\(24\!\cdots\!18\)\( T^{2} - 3329301676696184 p^{17} T^{3} + p^{34} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 13489402206504816 T + \)\(10\!\cdots\!46\)\( T^{2} - 13489402206504816 p^{17} T^{3} + p^{34} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 436589918136724 T - \)\(80\!\cdots\!94\)\( T^{2} - 436589918136724 p^{17} T^{3} + p^{34} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4376041565214880 T + \)\(25\!\cdots\!18\)\( T^{2} + 4376041565214880 p^{17} T^{3} + p^{34} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 39886442265612888 T + \)\(12\!\cdots\!18\)\( T^{2} - 39886442265612888 p^{17} T^{3} + p^{34} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6972184096107444 T + \)\(10\!\cdots\!98\)\( T^{2} - 6972184096107444 p^{17} T^{3} + p^{34} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 183569555712460996 T + \)\(20\!\cdots\!78\)\( T^{2} - 183569555712460996 p^{17} T^{3} + p^{34} T^{4} \) |
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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
−12.32739201811260367073228055188, −11.94158760461514423792350752794, −11.22793688512143638611924230614, −10.72879089397017797664359952931, −9.915772819888314610041045303501, −9.744262662011541290361122051726, −9.090578163731891307939857034830, −8.463392896836452904617787029549, −7.32251463155812670990489792956, −6.75975388710334132169229983259, −6.26178935564026436705405747691, −6.21046560951184380030983498753, −5.11700687023295049150719764603, −4.60219051004638793562348662949, −3.77847110890356776538504767418, −3.35687656285251231094317956707, −2.02102745330151391723133906960, −2.01772444893392570105589248812, −0.74339638033573545439891746291, −0.30654775242964367528555845085,
0.30654775242964367528555845085, 0.74339638033573545439891746291, 2.01772444893392570105589248812, 2.02102745330151391723133906960, 3.35687656285251231094317956707, 3.77847110890356776538504767418, 4.60219051004638793562348662949, 5.11700687023295049150719764603, 6.21046560951184380030983498753, 6.26178935564026436705405747691, 6.75975388710334132169229983259, 7.32251463155812670990489792956, 8.463392896836452904617787029549, 9.090578163731891307939857034830, 9.744262662011541290361122051726, 9.915772819888314610041045303501, 10.72879089397017797664359952931, 11.22793688512143638611924230614, 11.94158760461514423792350752794, 12.32739201811260367073228055188
