| L(s) = 1 | + 512·2-s + 1.30e3·3-s + 1.96e5·4-s + 6.69e5·6-s − 6.03e5·7-s + 6.71e7·8-s − 1.82e8·9-s − 4.71e8·11-s + 2.57e8·12-s + 1.54e9·13-s − 3.09e8·14-s + 2.14e10·16-s − 3.21e10·17-s − 9.32e10·18-s + 1.28e11·19-s − 7.89e8·21-s − 2.41e11·22-s − 6.50e11·23-s + 8.77e10·24-s + 7.89e11·26-s − 3.09e11·27-s − 1.18e11·28-s + 2.54e12·29-s + 7.83e12·31-s + 6.59e12·32-s − 6.16e11·33-s − 1.64e13·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.115·3-s + 3/2·4-s + 0.162·6-s − 0.0395·7-s + 1.41·8-s − 1.40·9-s − 0.663·11-s + 0.172·12-s + 0.524·13-s − 0.0559·14-s + 5/4·16-s − 1.11·17-s − 1.99·18-s + 1.73·19-s − 0.00455·21-s − 0.937·22-s − 1.73·23-s + 0.162·24-s + 0.741·26-s − 0.210·27-s − 0.0593·28-s + 0.944·29-s + 1.65·31-s + 1.06·32-s − 0.0763·33-s − 1.58·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(6.403555335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.403555335\) |
| \(L(\frac{19}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{8} T )^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 436 p T + 756194 p^{5} T^{2} - 436 p^{18} T^{3} + p^{34} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 603844 T - 3429560501298 p^{2} T^{2} + 603844 p^{17} T^{3} + p^{34} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 42861936 p T + 96764659462179986 p T^{2} + 42861936 p^{18} T^{3} + p^{34} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1541834228 T + 1332680700578493174 p T^{2} - 1541834228 p^{17} T^{3} + p^{34} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 32139900564 T + \)\(42\!\cdots\!78\)\( T^{2} + 32139900564 p^{17} T^{3} + p^{34} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 128672529400 T + \)\(11\!\cdots\!78\)\( T^{2} - 128672529400 p^{17} T^{3} + p^{34} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 650359859292 T + \)\(38\!\cdots\!22\)\( T^{2} + 650359859292 p^{17} T^{3} + p^{34} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2543054749980 T + \)\(13\!\cdots\!18\)\( T^{2} - 2543054749980 p^{17} T^{3} + p^{34} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7839407998744 T + \)\(44\!\cdots\!06\)\( T^{2} - 7839407998744 p^{17} T^{3} + p^{34} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 27805209097556 T + \)\(10\!\cdots\!18\)\( T^{2} - 27805209097556 p^{17} T^{3} + p^{34} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 37826364264156 T + \)\(40\!\cdots\!46\)\( T^{2} + 37826364264156 p^{17} T^{3} + p^{34} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 29630453926852 T + \)\(82\!\cdots\!62\)\( T^{2} + 29630453926852 p^{17} T^{3} + p^{34} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 220791583022004 T + \)\(41\!\cdots\!78\)\( T^{2} + 220791583022004 p^{17} T^{3} + p^{34} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1058306017600572 T + \)\(67\!\cdots\!22\)\( T^{2} + 1058306017600572 p^{17} T^{3} + p^{34} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1527820717235160 T + \)\(19\!\cdots\!38\)\( T^{2} - 1527820717235160 p^{17} T^{3} + p^{34} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 631738213122116 T + \)\(32\!\cdots\!06\)\( T^{2} + 631738213122116 p^{17} T^{3} + p^{34} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8843763493872596 T + \)\(38\!\cdots\!58\)\( T^{2} - 8843763493872596 p^{17} T^{3} + p^{34} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3331901660912376 T + \)\(57\!\cdots\!26\)\( T^{2} + 3331901660912376 p^{17} T^{3} + p^{34} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10026350574292028 T + \)\(11\!\cdots\!02\)\( T^{2} - 10026350574292028 p^{17} T^{3} + p^{34} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8263606842055120 T + \)\(37\!\cdots\!18\)\( T^{2} - 8263606842055120 p^{17} T^{3} + p^{34} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10689526892336988 T + \)\(86\!\cdots\!82\)\( T^{2} - 10689526892336988 p^{17} T^{3} + p^{34} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 55878412521904980 T + \)\(27\!\cdots\!58\)\( T^{2} - 55878412521904980 p^{17} T^{3} + p^{34} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 56106580363859756 T + \)\(41\!\cdots\!58\)\( T^{2} - 56106580363859756 p^{17} T^{3} + p^{34} T^{4} \) |
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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
−12.14408246824649772928674242612, −11.93105635291230589203532978700, −11.22128935150516565733068513006, −11.05357458634298925579467727706, −9.996175795106714503631668607248, −9.693302265160135260713847636824, −8.531727135416931760633872980117, −8.164552776480735781310844511595, −7.63860711998657507324612327635, −6.59158393078812550266118246882, −6.24541331901704517352361029020, −5.73410871726229684902996072750, −4.84705589806238709237154283888, −4.75009422741094693698854544419, −3.48853999647425812456738814804, −3.40614204075107887898696904609, −2.39960112557936649003855712084, −2.29806848867129728873189629275, −1.14779622756283865987264059177, −0.44182362925857320872466093936,
0.44182362925857320872466093936, 1.14779622756283865987264059177, 2.29806848867129728873189629275, 2.39960112557936649003855712084, 3.40614204075107887898696904609, 3.48853999647425812456738814804, 4.75009422741094693698854544419, 4.84705589806238709237154283888, 5.73410871726229684902996072750, 6.24541331901704517352361029020, 6.59158393078812550266118246882, 7.63860711998657507324612327635, 8.164552776480735781310844511595, 8.531727135416931760633872980117, 9.693302265160135260713847636824, 9.996175795106714503631668607248, 11.05357458634298925579467727706, 11.22128935150516565733068513006, 11.93105635291230589203532978700, 12.14408246824649772928674242612
