| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 3·7-s + 3·8-s + 3·9-s − 4·11-s + 2·12-s + 2·13-s + 3·14-s + 16-s + 5·17-s + 3·18-s − 3·19-s + 6·21-s − 4·22-s + 10·23-s + 6·24-s + 2·26-s + 4·27-s + 3·28-s + 11·29-s − 2·31-s − 32-s − 8·33-s + 5·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 1.13·7-s + 1.06·8-s + 9-s − 1.20·11-s + 0.577·12-s + 0.554·13-s + 0.801·14-s + 1/4·16-s + 1.21·17-s + 0.707·18-s − 0.688·19-s + 1.30·21-s − 0.852·22-s + 2.08·23-s + 1.22·24-s + 0.392·26-s + 0.769·27-s + 0.566·28-s + 2.04·29-s − 0.359·31-s − 0.176·32-s − 1.39·33-s + 0.857·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12425625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12425625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.89956182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.89956182\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 47 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 36 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 11 T + 84 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + T + 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 88 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 105 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 120 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 64 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 176 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 202 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.669681397964011109207828684804, −8.284695586302299561857484385063, −7.994729862267382647879033717713, −7.65873343707179697055732726414, −7.48991097791017129955641206943, −6.84160923472316500306942387236, −6.70445482231482709108752657254, −6.21893384929175497946332513386, −5.42324713103043363572157156872, −5.30331833841206127083425052647, −4.81549092131264996763420121182, −4.71924881410014352227092953463, −3.99853965729781136897422570655, −3.83798251862578031514072697662, −3.14254240021316603917510682609, −2.66742138424656439524716914777, −2.59615316542053371564676831553, −1.80577840753791379799851723050, −1.37195415941733273372058657826, −0.862694910913244328608913778349,
0.862694910913244328608913778349, 1.37195415941733273372058657826, 1.80577840753791379799851723050, 2.59615316542053371564676831553, 2.66742138424656439524716914777, 3.14254240021316603917510682609, 3.83798251862578031514072697662, 3.99853965729781136897422570655, 4.71924881410014352227092953463, 4.81549092131264996763420121182, 5.30331833841206127083425052647, 5.42324713103043363572157156872, 6.21893384929175497946332513386, 6.70445482231482709108752657254, 6.84160923472316500306942387236, 7.48991097791017129955641206943, 7.65873343707179697055732726414, 7.994729862267382647879033717713, 8.284695586302299561857484385063, 8.669681397964011109207828684804
