| L(s) = 1 | + 2·5-s − 2·7-s + 2·11-s + 2·13-s − 16·19-s − 6·23-s + 11·25-s − 10·29-s − 10·31-s − 4·35-s − 16·37-s − 14·41-s + 10·43-s + 2·47-s + 9·49-s + 16·53-s + 4·55-s − 14·59-s − 6·61-s + 4·65-s + 10·67-s + 8·71-s − 4·77-s − 22·79-s + 6·83-s + 32·89-s − 4·91-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s + 0.603·11-s + 0.554·13-s − 3.67·19-s − 1.25·23-s + 11/5·25-s − 1.85·29-s − 1.79·31-s − 0.676·35-s − 2.63·37-s − 2.18·41-s + 1.52·43-s + 0.291·47-s + 9/7·49-s + 2.19·53-s + 0.539·55-s − 1.82·59-s − 0.768·61-s + 0.496·65-s + 1.22·67-s + 0.949·71-s − 0.455·77-s − 2.47·79-s + 0.658·83-s + 3.39·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.114999018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.114999018\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 2 T - 5 T^{2} - 10 T^{3} + 4 T^{4} - 10 p T^{5} - 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 13 T^{2} + 10 T^{3} + 124 T^{4} + 10 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 2 T + T^{2} + 46 T^{3} - 212 T^{4} + 46 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T - 13 T^{2} + 18 T^{3} + 1044 T^{4} + 18 p T^{5} - 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 10 T + 41 T^{2} + 10 T^{3} - 260 T^{4} + 10 p T^{5} + 41 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 10 T + 19 T^{2} + 190 T^{3} + 2500 T^{4} + 190 p T^{5} + 19 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 14 T + 89 T^{2} + 350 T^{3} + 1732 T^{4} + 350 p T^{5} + 89 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 47 | $D_4\times C_2$ | \( 1 - 2 T - 37 T^{2} + 106 T^{3} - 716 T^{4} + 106 p T^{5} - 37 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 8 T + 98 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 14 T + 83 T^{2} - 70 T^{3} - 2276 T^{4} - 70 p T^{5} + 83 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 6 T - 71 T^{2} - 90 T^{3} + 5532 T^{4} - 90 p T^{5} - 71 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 10 T - 5 T^{2} + 290 T^{3} - 164 T^{4} + 290 p T^{5} - 5 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 22 T + 211 T^{2} + 2530 T^{3} + 30052 T^{4} + 2530 p T^{5} + 211 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 6 T - 133 T^{2} - 18 T^{3} + 18684 T^{4} - 18 p T^{5} - 133 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 16 T + 218 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 2 T - 167 T^{2} - 46 T^{3} + 19444 T^{4} - 46 p T^{5} - 167 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.49421285578656545044320834411, −6.43560970187427493951675803343, −6.26672527602408259595246852950, −5.95976360105852212278669026523, −5.80388060812864259137187820590, −5.77198414387048386542347719911, −5.34196964038995191305893000353, −5.26201605192546741285147422375, −4.78395875867142316656031008579, −4.68255053793144544708662919167, −4.60822594912482545536674037573, −3.94748068668391173135083523699, −3.91700146634609615867141711438, −3.78097871199629538270783210770, −3.68888130495642131227991742739, −3.26337406189175108265818819677, −3.05936857688656870062658009500, −2.56246810781075532655877075357, −2.30264260895303470375537844893, −2.04648328026271667662016722156, −2.01340010315726358138962090290, −1.51796729222640721659504144593, −1.46612916204850242837386191148, −0.61748662300169347866639523117, −0.23522431693810280523266761066,
0.23522431693810280523266761066, 0.61748662300169347866639523117, 1.46612916204850242837386191148, 1.51796729222640721659504144593, 2.01340010315726358138962090290, 2.04648328026271667662016722156, 2.30264260895303470375537844893, 2.56246810781075532655877075357, 3.05936857688656870062658009500, 3.26337406189175108265818819677, 3.68888130495642131227991742739, 3.78097871199629538270783210770, 3.91700146634609615867141711438, 3.94748068668391173135083523699, 4.60822594912482545536674037573, 4.68255053793144544708662919167, 4.78395875867142316656031008579, 5.26201605192546741285147422375, 5.34196964038995191305893000353, 5.77198414387048386542347719911, 5.80388060812864259137187820590, 5.95976360105852212278669026523, 6.26672527602408259595246852950, 6.43560970187427493951675803343, 6.49421285578656545044320834411
