| L(s) = 1 | − 2·3-s − 2·7-s + 6·11-s − 4·13-s − 2·17-s − 4·19-s + 4·21-s − 6·23-s + 2·25-s + 2·27-s − 2·31-s − 12·33-s − 16·37-s + 8·39-s − 12·41-s − 4·43-s − 8·49-s + 4·51-s − 12·53-s + 8·57-s − 12·59-s + 8·61-s − 16·67-s + 12·69-s − 6·71-s + 4·73-s − 4·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1.80·11-s − 1.10·13-s − 0.485·17-s − 0.917·19-s + 0.872·21-s − 1.25·23-s + 2/5·25-s + 0.384·27-s − 0.359·31-s − 2.08·33-s − 2.63·37-s + 1.28·39-s − 1.87·41-s − 0.609·43-s − 8/7·49-s + 0.560·51-s − 1.64·53-s + 1.05·57-s − 1.56·59-s + 1.02·61-s − 1.95·67-s + 1.44·69-s − 0.712·71-s + 0.468·73-s − 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 180 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 190 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.864522587100450236358327443498, −9.105116628896838512042105358906, −9.039212865467245279814440693438, −8.529340441713091781125251981079, −8.066704500498442809580938633487, −7.39132897354886857711951238396, −6.98774475090211207098656806155, −6.56909116194763809374856608461, −6.26279449209127142348219497505, −6.09622975587532286860639395999, −5.33614368183299603613454959117, −4.91033280036046228394644886774, −4.58295074127397763497136979886, −3.86375246473730332862260656487, −3.45962131514590313605230875255, −2.93683579973442792028204234987, −1.85416921761613474096660954846, −1.64459928959245029252102286296, 0, 0,
1.64459928959245029252102286296, 1.85416921761613474096660954846, 2.93683579973442792028204234987, 3.45962131514590313605230875255, 3.86375246473730332862260656487, 4.58295074127397763497136979886, 4.91033280036046228394644886774, 5.33614368183299603613454959117, 6.09622975587532286860639395999, 6.26279449209127142348219497505, 6.56909116194763809374856608461, 6.98774475090211207098656806155, 7.39132897354886857711951238396, 8.066704500498442809580938633487, 8.529340441713091781125251981079, 9.039212865467245279814440693438, 9.105116628896838512042105358906, 9.864522587100450236358327443498
