| L(s) = 1 | + 2·3-s + 12·7-s − 9-s − 8·11-s + 8·13-s − 6·17-s − 8·19-s + 24·21-s + 14·23-s − 2·27-s − 6·29-s − 16·33-s + 12·37-s + 16·39-s + 20·41-s + 10·43-s + 71·49-s − 12·51-s + 4·53-s − 16·57-s − 8·59-s + 18·61-s − 12·63-s − 28·67-s + 28·69-s − 8·71-s + 32·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 4.53·7-s − 1/3·9-s − 2.41·11-s + 2.21·13-s − 1.45·17-s − 1.83·19-s + 5.23·21-s + 2.91·23-s − 0.384·27-s − 1.11·29-s − 2.78·33-s + 1.97·37-s + 2.56·39-s + 3.12·41-s + 1.52·43-s + 71/7·49-s − 1.68·51-s + 0.549·53-s − 2.11·57-s − 1.04·59-s + 2.30·61-s − 1.51·63-s − 3.42·67-s + 3.37·69-s − 0.949·71-s + 3.74·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.46352899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.46352899\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 10 T^{3} + 16 T^{4} - 10 p T^{5} + 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 12 T + 73 T^{2} - 300 T^{3} + 912 T^{4} - 300 p T^{5} + 73 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 8 T + 41 T^{2} + 152 T^{3} + 532 T^{4} + 152 p T^{5} + 41 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 9 T^{2} - 6 p T^{3} - 688 T^{4} - 6 p^{2} T^{5} + 9 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 8 T + 41 T^{2} + 168 T^{3} + 644 T^{4} + 168 p T^{5} + 41 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 14 T + 53 T^{2} + 226 T^{3} - 2552 T^{4} + 226 p T^{5} + 53 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 1858 T^{4} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 109 T^{2} - 732 T^{3} + 4128 T^{4} - 732 p T^{5} + 109 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 20 T + 125 T^{2} + 160 T^{3} - 5156 T^{4} + 160 p T^{5} + 125 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 10 T + 29 T^{2} + 510 T^{3} - 5104 T^{4} + 510 p T^{5} + 29 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 2246 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 124 T^{3} + 1438 T^{4} - 124 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 8 T + 65 T^{2} - 8 p T^{3} - 52 p T^{4} - 8 p^{2} T^{5} + 65 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 9 T + 20 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 28 T + 457 T^{2} + 5404 T^{3} + 49912 T^{4} + 5404 p T^{5} + 457 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 8 T + 17 T^{2} - 688 T^{3} - 8612 T^{4} - 688 p T^{5} + 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 16 T + 162 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 28614 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 204 T^{2} + 21110 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 28 T + 197 T^{2} - 2408 T^{3} - 48788 T^{4} - 2408 p T^{5} + 197 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 12 T - 83 T^{2} + 396 T^{3} + 28152 T^{4} + 396 p T^{5} - 83 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.18482987284975876899492109511, −6.54453131215009765367528465809, −6.54441692001305465765941146047, −6.25693339878267570067354880501, −5.83752318110588018293643522136, −5.66556422304777833492521121597, −5.46463054955118823946635827560, −5.28589589346096637769712536790, −5.24857618019769452189515488495, −4.71121704520052090884210641254, −4.62650571098014960213627118732, −4.41780345790753144114569873167, −4.19025498037766928299365450632, −4.13188842430656132952744746955, −3.86347265416512004997542015074, −3.22067364091488973188222961943, −2.92296121702365548283170120511, −2.69833800552389329398593461356, −2.56806877926934140371085430867, −2.18378031236718945591933123760, −2.17361187695418897221548400929, −1.64874071714632276173618973925, −1.34540727165036218867727149516, −1.05163686308868157400305503841, −0.61568853857159655074151246325,
0.61568853857159655074151246325, 1.05163686308868157400305503841, 1.34540727165036218867727149516, 1.64874071714632276173618973925, 2.17361187695418897221548400929, 2.18378031236718945591933123760, 2.56806877926934140371085430867, 2.69833800552389329398593461356, 2.92296121702365548283170120511, 3.22067364091488973188222961943, 3.86347265416512004997542015074, 4.13188842430656132952744746955, 4.19025498037766928299365450632, 4.41780345790753144114569873167, 4.62650571098014960213627118732, 4.71121704520052090884210641254, 5.24857618019769452189515488495, 5.28589589346096637769712536790, 5.46463054955118823946635827560, 5.66556422304777833492521121597, 5.83752318110588018293643522136, 6.25693339878267570067354880501, 6.54441692001305465765941146047, 6.54453131215009765367528465809, 7.18482987284975876899492109511
