| L(s) = 1 | − 2·4-s − 2·9-s + 14·11-s + 3·16-s + 14·19-s + 12·29-s − 4·31-s + 4·36-s − 20·41-s − 28·44-s + 11·49-s + 12·59-s − 4·64-s + 2·71-s − 28·76-s + 30·79-s + 3·81-s − 6·89-s − 28·99-s − 2·101-s + 4·109-s − 24·116-s + 95·121-s + 8·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 4-s − 2/3·9-s + 4.22·11-s + 3/4·16-s + 3.21·19-s + 2.22·29-s − 0.718·31-s + 2/3·36-s − 3.12·41-s − 4.22·44-s + 11/7·49-s + 1.56·59-s − 1/2·64-s + 0.237·71-s − 3.21·76-s + 3.37·79-s + 1/3·81-s − 0.635·89-s − 2.81·99-s − 0.199·101-s + 0.383·109-s − 2.22·116-s + 8.63·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.30538589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.30538589\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 11 T^{2} + 120 T^{4} - 11 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $D_{4}$ | \( ( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 75 T^{2} + 2456 T^{4} - 75 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 32 T^{2} - 306 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 131 T^{2} + 7584 T^{4} - 131 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 7886 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 191 T^{2} + 14664 T^{4} - 191 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 152 T^{2} + 11454 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - T + 134 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 143 T^{2} + 15696 T^{4} - 143 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 15 T + 206 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 3 T + 172 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−5.76850325262519766482429540379, −5.69066009269952377787470003935, −5.50333201654178268739340122615, −5.22119626429661962061792738224, −5.06264363623654079150035824049, −4.95636753391204270425545821957, −4.75532876547109067257743985479, −4.32949750044530373399192064255, −4.28061554501289653605996187645, −4.06590189310268286276452468011, −3.84865571826920472773123663969, −3.63321904503824463639674719434, −3.48145439728809235187050913172, −3.44918238102332009410662083718, −2.94818821688529834835537399911, −2.88776746376828554023179361215, −2.86188752524180018574005077648, −2.20687777356438528072049199590, −1.80860740376077973947458704256, −1.65074247318675913164889480236, −1.65056518463477491577099012239, −1.12829709129633719385920323869, −0.836845891708379972901534893699, −0.66313467354204120915462330711, −0.63031658458938657219158967106,
0.63031658458938657219158967106, 0.66313467354204120915462330711, 0.836845891708379972901534893699, 1.12829709129633719385920323869, 1.65056518463477491577099012239, 1.65074247318675913164889480236, 1.80860740376077973947458704256, 2.20687777356438528072049199590, 2.86188752524180018574005077648, 2.88776746376828554023179361215, 2.94818821688529834835537399911, 3.44918238102332009410662083718, 3.48145439728809235187050913172, 3.63321904503824463639674719434, 3.84865571826920472773123663969, 4.06590189310268286276452468011, 4.28061554501289653605996187645, 4.32949750044530373399192064255, 4.75532876547109067257743985479, 4.95636753391204270425545821957, 5.06264363623654079150035824049, 5.22119626429661962061792738224, 5.50333201654178268739340122615, 5.69066009269952377787470003935, 5.76850325262519766482429540379
