| L(s) = 1 | − 4·11-s − 2·13-s − 4·17-s − 8·23-s − 2·25-s − 4·29-s + 8·31-s − 4·37-s − 16·41-s − 8·43-s − 12·47-s − 6·49-s + 4·53-s + 4·59-s + 4·61-s − 8·67-s + 4·71-s + 12·73-s − 4·83-s − 24·89-s − 4·97-s − 4·101-s − 16·103-s − 12·109-s − 12·113-s − 10·121-s + 127-s + ⋯ |
| L(s) = 1 | − 1.20·11-s − 0.554·13-s − 0.970·17-s − 1.66·23-s − 2/5·25-s − 0.742·29-s + 1.43·31-s − 0.657·37-s − 2.49·41-s − 1.21·43-s − 1.75·47-s − 6/7·49-s + 0.549·53-s + 0.520·59-s + 0.512·61-s − 0.977·67-s + 0.474·71-s + 1.40·73-s − 0.439·83-s − 2.54·89-s − 0.406·97-s − 0.398·101-s − 1.57·103-s − 1.14·109-s − 1.12·113-s − 0.909·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 314 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.004012180429742183752200119272, −8.372236965971839647406286301694, −8.184021155813878309610897240181, −8.157592139329312723069724392390, −7.52773608371643450728366162308, −6.95078715291688968303362776913, −6.57384898230558632176469021730, −6.52298667066576229102154261560, −5.65635968294031371748058215697, −5.41977881982989540036489871236, −4.93201872313107321656836048752, −4.67541977252246201873075662386, −3.92665575422878770169892297006, −3.71322838592949107005051803129, −2.89108547060620073612132694369, −2.63833872990398955983468995964, −1.87470028240941258930709038769, −1.58843629379767240162985800185, 0, 0,
1.58843629379767240162985800185, 1.87470028240941258930709038769, 2.63833872990398955983468995964, 2.89108547060620073612132694369, 3.71322838592949107005051803129, 3.92665575422878770169892297006, 4.67541977252246201873075662386, 4.93201872313107321656836048752, 5.41977881982989540036489871236, 5.65635968294031371748058215697, 6.52298667066576229102154261560, 6.57384898230558632176469021730, 6.95078715291688968303362776913, 7.52773608371643450728366162308, 8.157592139329312723069724392390, 8.184021155813878309610897240181, 8.372236965971839647406286301694, 9.004012180429742183752200119272
