L(s) = 1 | + 4·2-s − 18·3-s + 4·4-s + 48·5-s − 72·6-s − 72·7-s + 96·8-s + 243·9-s + 192·10-s + 596·11-s − 72·12-s − 288·14-s − 864·15-s − 176·16-s − 268·17-s + 972·18-s − 1.12e3·19-s + 192·20-s + 1.29e3·21-s + 2.38e3·22-s − 1.76e3·23-s − 1.72e3·24-s − 3.12e3·25-s − 2.91e3·27-s − 288·28-s − 7.61e3·29-s − 3.45e3·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/8·4-s + 0.858·5-s − 0.816·6-s − 0.555·7-s + 0.530·8-s + 9-s + 0.607·10-s + 1.48·11-s − 0.144·12-s − 0.392·14-s − 0.991·15-s − 0.171·16-s − 0.224·17-s + 0.707·18-s − 0.716·19-s + 0.107·20-s + 0.641·21-s + 1.05·22-s − 0.696·23-s − 0.612·24-s − 0.999·25-s − 0.769·27-s − 0.0694·28-s − 1.68·29-s − 0.701·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 13 | | \( 1 \) |
good | 2 | $D_{4}$ | \( 1 - p^{2} T + 3 p^{2} T^{2} - p^{7} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 48 T + 5426 T^{2} - 48 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 72 T + 28134 T^{2} + 72 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 596 T + 312122 T^{2} - 596 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 268 T + 2228454 T^{2} + 268 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1128 T + 4971870 T^{2} + 1128 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 1768 T + 12073598 T^{2} + 1768 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7612 T + 55112798 T^{2} + 7612 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4160 T + 24205318 T^{2} - 4160 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 17468 T + 184139086 T^{2} + 17468 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 28000 T + 412511706 T^{2} - 28000 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 24328 T + 8556274 p T^{2} + 24328 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18108 T + 278422754 T^{2} - 18108 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1420 T + 800695790 T^{2} - 1420 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6788 T - 153114342 T^{2} + 6788 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 37148 T + 1888667614 T^{2} + 37148 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 106112 T + 5372723950 T^{2} + 106112 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 30460 T + 2533993202 T^{2} - 30460 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 37620 T + 1766924310 T^{2} - 37620 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2160 T + 3629418462 T^{2} + 2160 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 207004 T + 18578057034 T^{2} + 207004 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 74136 T + 5425645898 T^{2} - 74136 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 121156 T + 20409714022 T^{2} - 121156 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.863302030749476803480155707614, −9.681350460710283426718968527101, −8.999875342845644065332302840664, −8.864533923441174308414769911720, −7.958110428218950066298393986630, −7.29317015150559974454288813868, −7.09778061339123488523031634835, −6.31868543525042773061129090488, −6.15805897978397588918029426574, −5.87874563454588432805188462257, −5.19472912728052422787679867881, −4.71806353639167747792155736564, −4.04054566568512454855862034541, −3.96852604300363477589993034008, −3.15492951618808295733910010739, −2.11202801021532689744941410831, −1.74283795751931457433697018750, −1.25617025032920634269471380917, 0, 0,
1.25617025032920634269471380917, 1.74283795751931457433697018750, 2.11202801021532689744941410831, 3.15492951618808295733910010739, 3.96852604300363477589993034008, 4.04054566568512454855862034541, 4.71806353639167747792155736564, 5.19472912728052422787679867881, 5.87874563454588432805188462257, 6.15805897978397588918029426574, 6.31868543525042773061129090488, 7.09778061339123488523031634835, 7.29317015150559974454288813868, 7.958110428218950066298393986630, 8.864533923441174308414769911720, 8.999875342845644065332302840664, 9.681350460710283426718968527101, 9.863302030749476803480155707614