| L(s) = 1 | + 3.93e4·3-s − 3.26e6·5-s − 2.70e7·7-s + 1.16e9·9-s + 1.06e10·11-s + 3.36e10·13-s − 1.28e11·15-s − 2.62e11·17-s + 4.35e11·19-s − 1.06e12·21-s + 6.01e12·23-s + 9.22e12·25-s + 3.05e13·27-s + 1.50e14·29-s + 8.64e13·31-s + 4.20e14·33-s + 8.82e13·35-s + 2.55e15·37-s + 1.32e15·39-s + 2.52e15·41-s + 4.59e14·43-s − 3.79e15·45-s − 9.73e15·47-s − 1.33e16·49-s − 1.03e16·51-s − 2.68e16·53-s − 3.48e16·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.748·5-s − 0.253·7-s + 9-s + 1.36·11-s + 0.880·13-s − 0.863·15-s − 0.537·17-s + 0.309·19-s − 0.292·21-s + 0.696·23-s + 0.483·25-s + 0.769·27-s + 1.92·29-s + 0.587·31-s + 1.57·33-s + 0.189·35-s + 3.23·37-s + 1.01·39-s + 1.20·41-s + 0.139·43-s − 0.748·45-s − 1.26·47-s − 1.17·49-s − 0.620·51-s − 1.11·53-s − 1.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(5.164574317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.164574317\) |
| \(L(\frac{21}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{9} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 3267108 T + 11616547094 p^{3} T^{2} + 3267108 p^{19} T^{3} + p^{38} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 27023984 T + 288236706876750 p^{2} T^{2} + 27023984 p^{19} T^{3} + p^{38} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 88280712 p^{2} T + 1053766276955164198 p^{2} T^{2} - 88280712 p^{21} T^{3} + p^{38} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 33667969900 T + \)\(24\!\cdots\!58\)\( p T^{2} - 33667969900 p^{19} T^{3} + p^{38} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 15454479132 p T + \)\(17\!\cdots\!10\)\( p^{2} T^{2} + 15454479132 p^{20} T^{3} + p^{38} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 22911464680 p T + \)\(90\!\cdots\!58\)\( T^{2} - 22911464680 p^{20} T^{3} + p^{38} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6013964563344 T + \)\(49\!\cdots\!58\)\( T^{2} - 6013964563344 p^{19} T^{3} + p^{38} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 150480267148332 T + \)\(15\!\cdots\!94\)\( T^{2} - 150480267148332 p^{19} T^{3} + p^{38} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 86418167256928 T - \)\(50\!\cdots\!62\)\( T^{2} - 86418167256928 p^{19} T^{3} + p^{38} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 69119983639564 p T + \)\(28\!\cdots\!02\)\( T^{2} - 69119983639564 p^{20} T^{3} + p^{38} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2521269021634740 T + \)\(63\!\cdots\!22\)\( T^{2} - 2521269021634740 p^{19} T^{3} + p^{38} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 459122670982024 T + \)\(16\!\cdots\!58\)\( T^{2} - 459122670982024 p^{19} T^{3} + p^{38} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9737516035986336 T + \)\(12\!\cdots\!90\)\( T^{2} + 9737516035986336 p^{19} T^{3} + p^{38} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 26851183352305092 T + \)\(13\!\cdots\!50\)\( T^{2} + 26851183352305092 p^{19} T^{3} + p^{38} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 113214081562642008 T + \)\(11\!\cdots\!94\)\( T^{2} + 113214081562642008 p^{19} T^{3} + p^{38} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 61288509973786676 T + \)\(14\!\cdots\!26\)\( T^{2} + 61288509973786676 p^{19} T^{3} + p^{38} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 457543185281514904 T + \)\(15\!\cdots\!10\)\( T^{2} - 457543185281514904 p^{19} T^{3} + p^{38} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 751970011770534960 T + \)\(37\!\cdots\!62\)\( T^{2} - 751970011770534960 p^{19} T^{3} + p^{38} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1019226905786569012 T + \)\(62\!\cdots\!10\)\( T^{2} - 1019226905786569012 p^{19} T^{3} + p^{38} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 601093912701153664 T + \)\(54\!\cdots\!62\)\( T^{2} - 601093912701153664 p^{19} T^{3} + p^{38} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1966104245797380936 T + \)\(60\!\cdots\!18\)\( T^{2} + 1966104245797380936 p^{19} T^{3} + p^{38} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8491700315755320300 T + \)\(38\!\cdots\!18\)\( T^{2} + 8491700315755320300 p^{19} T^{3} + p^{38} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 746042691667512772 T + \)\(43\!\cdots\!62\)\( T^{2} - 746042691667512772 p^{19} T^{3} + p^{38} T^{4} \) |
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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
−15.62164344509080877074104385044, −15.19426535579432410956147527527, −14.20893154034598802313486928285, −14.05953326410742733158910934155, −12.96027560437710064656444782943, −12.50143962273676612315896720316, −11.39414112459301864202272961485, −11.00948938241342596174105006787, −9.531605013729529301299697503098, −9.410780910208745748662419990785, −8.214169520317615543952988580251, −8.039421958348445779008322391608, −6.70705275055423707714254690683, −6.36436808471956355719556686558, −4.64067979460453200437395451069, −4.10378332790324489590619799077, −3.24412154110500059541335269942, −2.61017168555779300847244324211, −1.30847505637927072796519506378, −0.794838129351415917045789191823,
0.794838129351415917045789191823, 1.30847505637927072796519506378, 2.61017168555779300847244324211, 3.24412154110500059541335269942, 4.10378332790324489590619799077, 4.64067979460453200437395451069, 6.36436808471956355719556686558, 6.70705275055423707714254690683, 8.039421958348445779008322391608, 8.214169520317615543952988580251, 9.410780910208745748662419990785, 9.531605013729529301299697503098, 11.00948938241342596174105006787, 11.39414112459301864202272961485, 12.50143962273676612315896720316, 12.96027560437710064656444782943, 14.05953326410742733158910934155, 14.20893154034598802313486928285, 15.19426535579432410956147527527, 15.62164344509080877074104385044
