| L(s) = 1 | + 3·3-s + 3·5-s + 2·7-s + 3·9-s + 9·15-s + 6·17-s − 4·19-s + 6·21-s + 4·23-s + 5·25-s + 8·29-s + 7·31-s + 6·35-s + 37-s − 4·41-s + 24·43-s + 9·45-s + 8·47-s + 7·49-s + 18·51-s − 2·53-s − 12·57-s + 59-s − 4·61-s + 6·63-s − 20·67-s + 12·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s + 0.755·7-s + 9-s + 2.32·15-s + 1.45·17-s − 0.917·19-s + 1.30·21-s + 0.834·23-s + 25-s + 1.48·29-s + 1.25·31-s + 1.01·35-s + 0.164·37-s − 0.624·41-s + 3.65·43-s + 1.34·45-s + 1.16·47-s + 49-s + 2.52·51-s − 0.274·53-s − 1.58·57-s + 0.130·59-s − 0.512·61-s + 0.755·63-s − 2.44·67-s + 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.51369110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.51369110\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 3 | $C_4\times C_2$ | \( 1 - p T + 2 p T^{2} - p^{2} T^{3} + p^{2} T^{4} - p^{3} T^{5} + 2 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 - 3 T + 4 T^{2} + 3 T^{3} - 29 T^{4} + 3 p T^{5} + 4 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - 2 T - 3 T^{2} + 20 T^{3} - 19 T^{4} + 20 p T^{5} - 3 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 - 6 T + 19 T^{2} - 12 T^{3} - 251 T^{4} - 12 p T^{5} + 19 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 + 4 T - 3 T^{2} - 88 T^{3} - 295 T^{4} - 88 p T^{5} - 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 29 | $C_4\times C_2$ | \( 1 - 8 T + 35 T^{2} - 48 T^{3} - 631 T^{4} - 48 p T^{5} + 35 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 - 7 T + 18 T^{2} + 91 T^{3} - 1195 T^{4} + 91 p T^{5} + 18 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 - T - 36 T^{2} + 73 T^{3} + 1259 T^{4} + 73 p T^{5} - 36 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 + 4 T - 25 T^{2} - 264 T^{3} - 31 T^{4} - 264 p T^{5} - 25 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 - 8 T + 17 T^{2} + 240 T^{3} - 2719 T^{4} + 240 p T^{5} + 17 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 + 2 T - 49 T^{2} - 204 T^{3} + 2189 T^{4} - 204 p T^{5} - 49 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 - T - 58 T^{2} + 117 T^{3} + 3305 T^{4} + 117 p T^{5} - 58 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 + 4 T - 45 T^{2} - 424 T^{3} + 1049 T^{4} - 424 p T^{5} - 45 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 + 3 T - 62 T^{2} - 399 T^{3} + 3205 T^{4} - 399 p T^{5} - 62 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 + 16 T + 183 T^{2} + 1760 T^{3} + 14801 T^{4} + 1760 p T^{5} + 183 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 + 2 T - 75 T^{2} - 308 T^{3} + 5309 T^{4} - 308 p T^{5} - 75 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 - 2 T - 79 T^{2} + 324 T^{3} + 5909 T^{4} + 324 p T^{5} - 79 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 - 7 T - 48 T^{2} + 1015 T^{3} - 2449 T^{4} + 1015 p T^{5} - 48 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.38955696353819133957189249711, −7.16972612596115452434938281346, −6.65791449465686385792090293700, −6.34189716724662700209830897321, −6.25313016557918797914975367192, −6.07348225004211130398909240882, −6.04858726616496073563299531397, −5.44280248698738709613590845828, −5.30238651475877726397394844620, −5.23852238894993523173212241603, −4.85309193790736958329202121085, −4.42623737701620718282474878193, −4.40568529117200327084815735266, −4.05594038645738552165813442635, −3.99016777157255475874642830355, −3.24044442048715294168693551168, −3.14289928745319323238861369969, −2.99662220801458543063476476970, −2.82503042589470288334857758274, −2.23825032858746089239386070766, −2.16687896525309842154675076157, −2.07315177572463068799249620681, −1.35434653629469957695064567135, −1.01710887231131229252452327497, −0.838495965645129425409380650378,
0.838495965645129425409380650378, 1.01710887231131229252452327497, 1.35434653629469957695064567135, 2.07315177572463068799249620681, 2.16687896525309842154675076157, 2.23825032858746089239386070766, 2.82503042589470288334857758274, 2.99662220801458543063476476970, 3.14289928745319323238861369969, 3.24044442048715294168693551168, 3.99016777157255475874642830355, 4.05594038645738552165813442635, 4.40568529117200327084815735266, 4.42623737701620718282474878193, 4.85309193790736958329202121085, 5.23852238894993523173212241603, 5.30238651475877726397394844620, 5.44280248698738709613590845828, 6.04858726616496073563299531397, 6.07348225004211130398909240882, 6.25313016557918797914975367192, 6.34189716724662700209830897321, 6.65791449465686385792090293700, 7.16972612596115452434938281346, 7.38955696353819133957189249711
