| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s − 2·7-s − 4·8-s − 4·10-s + 9·11-s − 5·13-s + 4·14-s + 5·16-s + 4·17-s + 9·19-s + 6·20-s − 18·22-s + 7·23-s − 2·25-s + 10·26-s − 6·28-s + 15·29-s + 2·31-s − 6·32-s − 8·34-s − 4·35-s − 4·37-s − 18·38-s − 8·40-s + 4·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.894·5-s − 0.755·7-s − 1.41·8-s − 1.26·10-s + 2.71·11-s − 1.38·13-s + 1.06·14-s + 5/4·16-s + 0.970·17-s + 2.06·19-s + 1.34·20-s − 3.83·22-s + 1.45·23-s − 2/5·25-s + 1.96·26-s − 1.13·28-s + 2.78·29-s + 0.359·31-s − 1.06·32-s − 1.37·34-s − 0.676·35-s − 0.657·37-s − 2.91·38-s − 1.26·40-s + 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55264356 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55264356 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.842447189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.842447189\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 59 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 9 T + 41 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 21 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 9 T + 47 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 47 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 15 T + 113 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 3 T - 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 73 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T + 45 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 22 T + 274 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 290 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 7 T + p T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3 T + 185 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−8.139181243804712772512707268857, −7.69059718454582682952561272451, −7.34114473410668637573269583866, −6.88560310608813111053490821390, −6.79442664331373670471451832470, −6.67872065451257891077548702143, −5.99497586601127500903600975649, −5.90160239720400117133440808233, −5.26978079124217454562178748167, −5.15710892198458729851262361387, −4.40603047166728741178583442293, −4.18811158469793927795477253951, −3.47333557862096622747967088109, −3.18465293703152096342756447007, −2.69803710634373312177474425615, −2.66194118913418140765985950993, −1.57436215308314282384511078641, −1.52208594512164415383041032332, −0.975531080358062073749022312112, −0.63276370369498077598758320683,
0.63276370369498077598758320683, 0.975531080358062073749022312112, 1.52208594512164415383041032332, 1.57436215308314282384511078641, 2.66194118913418140765985950993, 2.69803710634373312177474425615, 3.18465293703152096342756447007, 3.47333557862096622747967088109, 4.18811158469793927795477253951, 4.40603047166728741178583442293, 5.15710892198458729851262361387, 5.26978079124217454562178748167, 5.90160239720400117133440808233, 5.99497586601127500903600975649, 6.67872065451257891077548702143, 6.79442664331373670471451832470, 6.88560310608813111053490821390, 7.34114473410668637573269583866, 7.69059718454582682952561272451, 8.139181243804712772512707268857
