| L(s) = 1 | + 4·5-s + 10·25-s + 24·37-s − 16·41-s + 16·43-s − 8·47-s + 6·49-s + 24·59-s − 40·83-s + 8·89-s + 16·101-s + 40·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 96·185-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 2·25-s + 3.94·37-s − 2.49·41-s + 2.43·43-s − 1.16·47-s + 6/7·49-s + 3.12·59-s − 4.39·83-s + 0.847·89-s + 1.59·101-s + 3.63·121-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 7.05·185-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.272808350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.272808350\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 454 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1318 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 36 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^3$ | \( 1 - 1198 T^{4} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4342 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 8434 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 11638 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 200 T^{2} + 19762 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 7306 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 16426 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.86831662969288081170156018690, −6.73232874405812144170233245513, −6.31369365270067182528244615048, −6.23415414793437554856738153506, −6.07381118378861489782110135714, −5.86991191315157611264339262095, −5.51015636560288849467627020334, −5.46248208804718704791377507115, −5.38479831534244802993383833524, −4.77793569126490184791762728384, −4.64917721238195106306579365934, −4.57889551732906068067147480938, −4.29166314126879988824562879948, −3.91027954257006414471065445283, −3.56784680945003752827092825017, −3.50994937761847695057461715331, −3.06615206883710059561771059065, −2.68432893105141456885263690152, −2.39954350126897976849378557097, −2.33425079451305297894494726039, −2.24338965360079217020875269931, −1.49876511625238051038305719322, −1.28949772123454285092429346013, −1.04769359942725552345564523785, −0.46768418022152930161669348291,
0.46768418022152930161669348291, 1.04769359942725552345564523785, 1.28949772123454285092429346013, 1.49876511625238051038305719322, 2.24338965360079217020875269931, 2.33425079451305297894494726039, 2.39954350126897976849378557097, 2.68432893105141456885263690152, 3.06615206883710059561771059065, 3.50994937761847695057461715331, 3.56784680945003752827092825017, 3.91027954257006414471065445283, 4.29166314126879988824562879948, 4.57889551732906068067147480938, 4.64917721238195106306579365934, 4.77793569126490184791762728384, 5.38479831534244802993383833524, 5.46248208804718704791377507115, 5.51015636560288849467627020334, 5.86991191315157611264339262095, 6.07381118378861489782110135714, 6.23415414793437554856738153506, 6.31369365270067182528244615048, 6.73232874405812144170233245513, 6.86831662969288081170156018690
