| L(s) = 1 | − 2·3-s + 10·5-s + 16·7-s − 45·9-s − 90·11-s + 50·13-s − 20·15-s − 44·17-s + 108·19-s − 32·21-s + 28·23-s + 41·25-s + 134·27-s − 58·29-s − 66·31-s + 180·33-s + 160·35-s − 40·37-s − 100·39-s + 304·41-s − 130·43-s − 450·45-s + 514·47-s − 110·49-s + 88·51-s + 958·53-s − 900·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 0.894·5-s + 0.863·7-s − 5/3·9-s − 2.46·11-s + 1.06·13-s − 0.344·15-s − 0.627·17-s + 1.30·19-s − 0.332·21-s + 0.253·23-s + 0.327·25-s + 0.955·27-s − 0.371·29-s − 0.382·31-s + 0.949·33-s + 0.772·35-s − 0.177·37-s − 0.410·39-s + 1.15·41-s − 0.461·43-s − 1.49·45-s + 1.59·47-s − 0.320·49-s + 0.241·51-s + 2.48·53-s − 2.20·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3444736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3444736 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 49 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 p T + 59 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 16 T + 366 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 90 T + 4633 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 50 T + 4635 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 44 T + 8774 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 108 T + 16538 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 28 T - 11974 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 66 T - 10615 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 40 T + 101610 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 304 T + 122546 T^{2} - 304 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 130 T + 146385 T^{2} + 130 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 514 T + 214889 T^{2} - 514 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 958 T + 510971 T^{2} - 958 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 180 T + 43858 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1028 T + 717294 T^{2} + 1028 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 912 T + 807062 T^{2} + 912 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 796 T + 867290 T^{2} + 796 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 856 T + 775362 T^{2} + 856 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 318 T + 229433 T^{2} - 318 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1828 T + 1970306 T^{2} + 1828 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 944 T + 1601618 T^{2} + 944 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 368 T + 1799202 T^{2} - 368 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.649432880170160118005311612871, −8.437023108448333657907754471048, −7.84787548819898435838716734112, −7.53746659577532390884028573368, −7.29537203687912585804599637551, −6.65184562723849352527365855206, −5.87798411422825086134509238860, −5.72937021191881752516631433901, −5.66656675273857799520378684228, −5.26876667714268794826931497558, −4.61766087645681100846972436610, −4.41759788704117786424248447375, −3.45634121605628699592929926000, −3.03260741226006218298931538634, −2.52601432865360627852857434090, −2.40259961699184181315462283374, −1.49205509143718277256381854908, −1.06507724136839460011055844376, 0, 0,
1.06507724136839460011055844376, 1.49205509143718277256381854908, 2.40259961699184181315462283374, 2.52601432865360627852857434090, 3.03260741226006218298931538634, 3.45634121605628699592929926000, 4.41759788704117786424248447375, 4.61766087645681100846972436610, 5.26876667714268794826931497558, 5.66656675273857799520378684228, 5.72937021191881752516631433901, 5.87798411422825086134509238860, 6.65184562723849352527365855206, 7.29537203687912585804599637551, 7.53746659577532390884028573368, 7.84787548819898435838716734112, 8.437023108448333657907754471048, 8.649432880170160118005311612871
