| L(s) = 1 | − 8·2-s + 20·3-s + 48·4-s − 160·6-s − 98·7-s − 256·8-s + 130·9-s − 1.07e3·11-s + 960·12-s + 736·13-s + 784·14-s + 1.28e3·16-s − 1.90e3·17-s − 1.04e3·18-s + 828·19-s − 1.96e3·21-s + 8.56e3·22-s + 1.65e3·23-s − 5.12e3·24-s − 5.88e3·26-s + 2.06e3·27-s − 4.70e3·28-s − 2.24e3·29-s + 288·31-s − 6.14e3·32-s − 2.14e4·33-s + 1.52e4·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.28·3-s + 3/2·4-s − 1.81·6-s − 0.755·7-s − 1.41·8-s + 0.534·9-s − 2.66·11-s + 1.92·12-s + 1.20·13-s + 1.06·14-s + 5/4·16-s − 1.59·17-s − 0.756·18-s + 0.526·19-s − 0.969·21-s + 3.77·22-s + 0.651·23-s − 1.81·24-s − 1.70·26-s + 0.543·27-s − 1.13·28-s − 0.495·29-s + 0.0538·31-s − 1.06·32-s − 3.42·33-s + 2.25·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.015136344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.015136344\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 20 T + 10 p^{3} T^{2} - 20 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 1070 T + 605483 T^{2} + 1070 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 736 T + 612254 T^{2} - 736 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 112 p T + 2956018 T^{2} + 112 p^{6} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 828 T + 1016858 T^{2} - 828 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1654 T + 13124011 T^{2} - 1654 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2246 T + 41959843 T^{2} + 2246 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 288 T - 336926 T^{2} - 288 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8986 T + 110811363 T^{2} - 8986 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 11228 T + 235355354 T^{2} - 11228 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3402 T + 111010963 T^{2} - 3402 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 15572 T + 515625986 T^{2} - 15572 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16188 T + 615511758 T^{2} + 16188 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 92928 T + 60239502 p T^{2} + 92928 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 15172 T + 1660964642 T^{2} - 15172 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 78966 T + 3383134403 T^{2} - 78966 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20822 T + 2040843979 T^{2} + 20822 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 72884 T + 73693322 p T^{2} - 72884 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 72106 T + 6285386043 T^{2} + 72106 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 47400 T + 1662784410 T^{2} + 47400 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9920 T + 10081782998 T^{2} + 9920 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 26792 T + 6506522934 T^{2} + 26792 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.77660990074949290357877825893, −10.37470355691241047890832301946, −9.797621027459837193170485978171, −9.427141635501463264899931472946, −8.839723220370905876650185364802, −8.776625447618634591824818280420, −8.001903636407612177666799606538, −7.938149967772523902785291675540, −7.34348175183006946809198209342, −6.90064993649223603580672891589, −6.09257164445169759043479253468, −5.86432042534986802348517008795, −4.97687701048653956017438682014, −4.31299134906452986014163043072, −3.23069299962421065946795879124, −3.08966169579863965748199309705, −2.39513923449969653841464586965, −2.12672100287617238692924190592, −0.996066714943728824782895902483, −0.33790409625047817465841841411,
0.33790409625047817465841841411, 0.996066714943728824782895902483, 2.12672100287617238692924190592, 2.39513923449969653841464586965, 3.08966169579863965748199309705, 3.23069299962421065946795879124, 4.31299134906452986014163043072, 4.97687701048653956017438682014, 5.86432042534986802348517008795, 6.09257164445169759043479253468, 6.90064993649223603580672891589, 7.34348175183006946809198209342, 7.938149967772523902785291675540, 8.001903636407612177666799606538, 8.776625447618634591824818280420, 8.839723220370905876650185364802, 9.427141635501463264899931472946, 9.797621027459837193170485978171, 10.37470355691241047890832301946, 10.77660990074949290357877825893
