| L(s) = 1 | − 4-s + 4·5-s − 3·16-s + 12·17-s − 4·20-s + 6·25-s − 4·37-s + 4·41-s + 8·43-s − 8·47-s + 8·59-s + 7·64-s − 8·67-s − 12·68-s − 12·80-s + 48·85-s + 4·89-s − 6·100-s − 4·101-s + 28·109-s − 10·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 4·148-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.78·5-s − 3/4·16-s + 2.91·17-s − 0.894·20-s + 6/5·25-s − 0.657·37-s + 0.624·41-s + 1.21·43-s − 1.16·47-s + 1.04·59-s + 7/8·64-s − 0.977·67-s − 1.45·68-s − 1.34·80-s + 5.20·85-s + 0.423·89-s − 3/5·100-s − 0.398·101-s + 2.68·109-s − 0.909·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.328·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.255104480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.255104480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.241491572692316647575175944773, −8.784745183007253920781835411734, −8.211884608587383091651694603497, −7.64698318427314520306544813818, −7.27662354948317530374607711988, −6.52544951464709992016558039521, −6.06220614086007746387488529164, −5.61427078423598601723842316013, −5.29337353624810588082433847695, −4.75046263301563460713424867408, −3.92727170724649136904494514058, −3.29297851348166257811974706170, −2.59767463997901545270087843660, −1.83591738529574353025872323897, −1.05946729006000064735378091667,
1.05946729006000064735378091667, 1.83591738529574353025872323897, 2.59767463997901545270087843660, 3.29297851348166257811974706170, 3.92727170724649136904494514058, 4.75046263301563460713424867408, 5.29337353624810588082433847695, 5.61427078423598601723842316013, 6.06220614086007746387488529164, 6.52544951464709992016558039521, 7.27662354948317530374607711988, 7.64698318427314520306544813818, 8.211884608587383091651694603497, 8.784745183007253920781835411734, 9.241491572692316647575175944773
