| L(s) = 1 | − 512·2-s − 1.30e3·3-s + 1.96e5·4-s + 7.81e5·5-s + 6.69e5·6-s + 6.03e5·7-s − 6.71e7·8-s − 1.82e8·9-s − 4.00e8·10-s − 4.71e8·11-s − 2.57e8·12-s − 1.54e9·13-s − 3.09e8·14-s − 1.02e9·15-s + 2.14e10·16-s + 3.21e10·17-s + 9.32e10·18-s + 1.28e11·19-s + 1.53e11·20-s − 7.89e8·21-s + 2.41e11·22-s + 6.50e11·23-s + 8.77e10·24-s + 4.57e11·25-s + 7.89e11·26-s + 3.09e11·27-s + 1.18e11·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.115·3-s + 3/2·4-s + 0.894·5-s + 0.162·6-s + 0.0395·7-s − 1.41·8-s − 1.40·9-s − 1.26·10-s − 0.663·11-s − 0.172·12-s − 0.524·13-s − 0.0559·14-s − 0.102·15-s + 5/4·16-s + 1.11·17-s + 1.99·18-s + 1.73·19-s + 1.34·20-s − 0.00455·21-s + 0.937·22-s + 1.73·23-s + 0.162·24-s + 3/5·25-s + 0.741·26-s + 0.210·27-s + 0.0593·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.376771223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376771223\) |
| \(L(\frac{19}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{8} T )^{2} \) |
| 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
−16.91331647796161257372432449623, −16.71190426901195853340748588141, −15.70921941218575105644057520023, −14.92114101550407849727475476248, −14.06847169365456172570887139250, −13.43605969509634101210197496410, −11.90456567292496176224001549305, −11.86821136002649854971542806552, −10.37125833178106857491133091068, −10.32654301785782703697713516185, −9.156191399489473612719585899751, −8.670268548833463173601824657424, −7.66158257804266450132459066986, −6.89507437872306717036684652840, −5.66588688571795440059821513094, −5.26730866841669086707818893707, −2.91054587554933684561382951429, −2.72388181303130273758290534286, −1.25154400183761327838886927625, −0.62687264407191403541721810399,
0.62687264407191403541721810399, 1.25154400183761327838886927625, 2.72388181303130273758290534286, 2.91054587554933684561382951429, 5.26730866841669086707818893707, 5.66588688571795440059821513094, 6.89507437872306717036684652840, 7.66158257804266450132459066986, 8.670268548833463173601824657424, 9.156191399489473612719585899751, 10.32654301785782703697713516185, 10.37125833178106857491133091068, 11.86821136002649854971542806552, 11.90456567292496176224001549305, 13.43605969509634101210197496410, 14.06847169365456172570887139250, 14.92114101550407849727475476248, 15.70921941218575105644057520023, 16.71190426901195853340748588141, 16.91331647796161257372432449623
