| L(s) = 1 | + 594·2-s + 1.31e4·3-s + 1.33e5·4-s + 3.82e5·5-s + 7.79e6·6-s + 2.44e7·7-s + 2.69e7·8-s + 1.29e8·9-s + 2.27e8·10-s − 9.87e8·11-s + 1.75e9·12-s − 2.51e9·13-s + 1.45e10·14-s + 5.02e9·15-s + 1.58e10·16-s − 3.43e10·17-s + 7.67e10·18-s + 8.00e10·19-s + 5.11e10·20-s + 3.21e11·21-s − 5.86e11·22-s + 2.97e11·23-s + 3.54e11·24-s + 2.08e11·25-s − 1.49e12·26-s + 1.12e12·27-s + 3.26e12·28-s + ⋯ |
| L(s) = 1 | + 1.64·2-s + 1.15·3-s + 1.01·4-s + 0.438·5-s + 1.89·6-s + 1.60·7-s + 0.568·8-s + 9-s + 0.719·10-s − 1.38·11-s + 1.17·12-s − 0.856·13-s + 2.63·14-s + 0.506·15-s + 0.920·16-s − 1.19·17-s + 1.64·18-s + 1.08·19-s + 0.446·20-s + 1.85·21-s − 2.27·22-s + 0.791·23-s + 0.656·24-s + 0.273·25-s − 1.40·26-s + 0.769·27-s + 1.63·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(7.756432725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.756432725\) |
| \(L(\frac{19}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{8} T )^{2} \) |
| 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
−22.41705788410736049892210111130, −21.85999246420172938987640576798, −20.81049141364221642616808122999, −20.77973404631493090688209368255, −19.54454648704164377391957063196, −18.30639158020936681052018018951, −17.48207854395165821559506257572, −15.84042662563218817911405150332, −14.78912087820957412594154867832, −14.41422208150725484529006746713, −13.40368287206343907489735818945, −13.05073225793903038042554351315, −11.47466914490790494379001657764, −10.10098342325231373405888638932, −8.435139991456860827957145782434, −7.39551238701347746065126219882, −5.10016691847058451939289308147, −4.66470067397013088175077662639, −2.97393553239677131736204792795, −1.79667147775373923209916612009,
1.79667147775373923209916612009, 2.97393553239677131736204792795, 4.66470067397013088175077662639, 5.10016691847058451939289308147, 7.39551238701347746065126219882, 8.435139991456860827957145782434, 10.10098342325231373405888638932, 11.47466914490790494379001657764, 13.05073225793903038042554351315, 13.40368287206343907489735818945, 14.41422208150725484529006746713, 14.78912087820957412594154867832, 15.84042662563218817911405150332, 17.48207854395165821559506257572, 18.30639158020936681052018018951, 19.54454648704164377391957063196, 20.77973404631493090688209368255, 20.81049141364221642616808122999, 21.85999246420172938987640576798, 22.41705788410736049892210111130
