| L(s) = 1 | − 2·2-s + 3·4-s + 2·7-s − 4·8-s + 3·11-s − 2·13-s − 4·14-s + 5·16-s + 17-s + 7·19-s − 6·22-s − 3·23-s + 4·26-s + 6·28-s + 3·29-s + 7·31-s − 6·32-s − 2·34-s − 2·37-s − 14·38-s + 11·41-s + 14·43-s + 9·44-s + 6·46-s − 11·47-s + 3·49-s − 6·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s + 0.904·11-s − 0.554·13-s − 1.06·14-s + 5/4·16-s + 0.242·17-s + 1.60·19-s − 1.27·22-s − 0.625·23-s + 0.784·26-s + 1.13·28-s + 0.557·29-s + 1.25·31-s − 1.06·32-s − 0.342·34-s − 0.328·37-s − 2.27·38-s + 1.71·41-s + 2.13·43-s + 1.35·44-s + 0.884·46-s − 1.60·47-s + 3/7·49-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.080946498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.080946498\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - 3 T - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 36 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 50 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T - 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 11 T + 88 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 100 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - T + 118 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5 T + 128 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 25 T + 298 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−7.75945440961521878635015619901, −7.55328977730697777566663347456, −7.30747627034380874196053803739, −7.19170628569305787856895060116, −6.42508496443011791599354783734, −6.37874294761802079250141730761, −5.84179782729380627942000723841, −5.74662113277053634053109974186, −5.07019661601460422570935549524, −4.82157164955359703962345612338, −4.39670973595914339416190351526, −4.03948931634693608489759938734, −3.42169317763331040942022624053, −3.20299263668525711166953292924, −2.53580149543667072576251740225, −2.42808887357122607320666976207, −1.77375558149517845010258689688, −1.28035722352257182438114929080, −1.01260825345890657038299204786, −0.48814950526400789213457592474,
0.48814950526400789213457592474, 1.01260825345890657038299204786, 1.28035722352257182438114929080, 1.77375558149517845010258689688, 2.42808887357122607320666976207, 2.53580149543667072576251740225, 3.20299263668525711166953292924, 3.42169317763331040942022624053, 4.03948931634693608489759938734, 4.39670973595914339416190351526, 4.82157164955359703962345612338, 5.07019661601460422570935549524, 5.74662113277053634053109974186, 5.84179782729380627942000723841, 6.37874294761802079250141730761, 6.42508496443011791599354783734, 7.19170628569305787856895060116, 7.30747627034380874196053803739, 7.55328977730697777566663347456, 7.75945440961521878635015619901
