| L(s) = 1 | − 594·2-s + 1.33e5·4-s − 3.82e5·5-s + 2.44e7·7-s − 2.69e7·8-s + 2.27e8·10-s + 9.87e8·11-s − 2.51e9·13-s − 1.45e10·14-s + 1.58e10·16-s + 3.43e10·17-s + 8.00e10·19-s − 5.11e10·20-s − 5.86e11·22-s − 2.97e11·23-s + 2.08e11·25-s + 1.49e12·26-s + 3.26e12·28-s + 4.70e11·29-s + 3.40e12·31-s − 3.67e12·32-s − 2.03e13·34-s − 9.36e12·35-s + 1.06e13·37-s − 4.75e13·38-s + 1.03e13·40-s + 1.13e14·41-s + ⋯ |
| L(s) = 1 | − 1.64·2-s + 1.01·4-s − 0.438·5-s + 1.60·7-s − 0.568·8-s + 0.719·10-s + 1.38·11-s − 0.856·13-s − 2.63·14-s + 0.920·16-s + 1.19·17-s + 1.08·19-s − 0.446·20-s − 2.27·22-s − 0.791·23-s + 0.273·25-s + 1.40·26-s + 1.63·28-s + 0.174·29-s + 0.716·31-s − 0.590·32-s − 1.95·34-s − 0.703·35-s + 0.498·37-s − 1.77·38-s + 0.249·40-s + 2.21·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.395636187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.395636187\) |
| \(L(\frac{19}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + 297 p T + 6851 p^{5} T^{2} + 297 p^{18} T^{3} + p^{34} T^{4} \) |
| 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
−17.34414913779403351298239903316, −17.06364736654345400005959147632, −16.19446765847558636690099321161, −15.20288176622265006953580310497, −14.25549278157284375658471260834, −14.25263409098795832740525477833, −12.29154156173626048938470904555, −11.90020173081925787600603400079, −11.16130012251580709148360934338, −10.12014993243829902696537758119, −9.407295597891158730038533639508, −8.763106050808000247910298981740, −7.69529648333991374460729900145, −7.65758147541589099429127755423, −6.04832504035247431722767009581, −4.87368275945166556478348150124, −3.79787175609358135891789349035, −2.33131230755794192488502596700, −1.00561375945946341752624147608, −0.861260196811737651451318436120,
0.861260196811737651451318436120, 1.00561375945946341752624147608, 2.33131230755794192488502596700, 3.79787175609358135891789349035, 4.87368275945166556478348150124, 6.04832504035247431722767009581, 7.65758147541589099429127755423, 7.69529648333991374460729900145, 8.763106050808000247910298981740, 9.407295597891158730038533639508, 10.12014993243829902696537758119, 11.16130012251580709148360934338, 11.90020173081925787600603400079, 12.29154156173626048938470904555, 14.25263409098795832740525477833, 14.25549278157284375658471260834, 15.20288176622265006953580310497, 16.19446765847558636690099321161, 17.06364736654345400005959147632, 17.34414913779403351298239903316
