| L(s) = 1 | + 2·2-s + 4-s − 2·8-s + 3·9-s + 20·13-s − 4·16-s − 2·17-s + 6·18-s − 8·19-s − 10·25-s + 40·26-s − 2·32-s − 4·34-s + 3·36-s − 16·38-s + 32·43-s − 8·47-s − 2·49-s − 20·50-s + 20·52-s + 2·53-s − 24·59-s + 3·64-s + 32·67-s − 2·68-s − 6·72-s − 8·76-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.707·8-s + 9-s + 5.54·13-s − 16-s − 0.485·17-s + 1.41·18-s − 1.83·19-s − 2·25-s + 7.84·26-s − 0.353·32-s − 0.685·34-s + 1/2·36-s − 2.59·38-s + 4.87·43-s − 1.16·47-s − 2/7·49-s − 2.82·50-s + 2.77·52-s + 0.274·53-s − 3.12·59-s + 3/8·64-s + 3.90·67-s − 0.242·68-s − 0.707·72-s − 0.917·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.223164360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.223164360\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 5 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 59 T^{2} + p^{2} T^{4} ) \) |
| 37 | $C_2^3$ | \( 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 141 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 + 82 T^{2} + 1395 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 + 109 T^{2} + 5640 T^{4} + 109 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.840872255957439781817503957091, −8.564029696573533914418462573057, −8.308795883507136527433395593556, −8.168956086869568354640259688969, −8.020808056541671427869969667161, −7.54996769182271527692556209778, −7.03205899264091152436941442234, −6.95067880007648265560145080139, −6.40428611488213842166283134958, −6.38956897147667516533394771779, −5.98621119760161825965706189649, −5.86475833711548308583325196366, −5.73384178287771844887039889638, −5.53571914412022889265023203023, −4.76011602936817966651137700162, −4.42805155635050577051588752473, −4.13547027365039295748053769346, −3.97350140959636375341568508133, −3.80951710871072641624043892974, −3.76605421545594811879256021514, −2.88194820610196502573256478848, −2.84326857448387940484128249192, −1.87072485956510633802368904664, −1.59337003500745940107620069592, −1.04546580915508503420903568006,
1.04546580915508503420903568006, 1.59337003500745940107620069592, 1.87072485956510633802368904664, 2.84326857448387940484128249192, 2.88194820610196502573256478848, 3.76605421545594811879256021514, 3.80951710871072641624043892974, 3.97350140959636375341568508133, 4.13547027365039295748053769346, 4.42805155635050577051588752473, 4.76011602936817966651137700162, 5.53571914412022889265023203023, 5.73384178287771844887039889638, 5.86475833711548308583325196366, 5.98621119760161825965706189649, 6.38956897147667516533394771779, 6.40428611488213842166283134958, 6.95067880007648265560145080139, 7.03205899264091152436941442234, 7.54996769182271527692556209778, 8.020808056541671427869969667161, 8.168956086869568354640259688969, 8.308795883507136527433395593556, 8.564029696573533914418462573057, 8.840872255957439781817503957091
