| L(s) = 1 | − 2·4-s − 7-s − 4·9-s − 11-s + 5·13-s + 5·17-s − 3·19-s + 5·23-s − 2·25-s + 2·28-s − 5·29-s − 2·31-s + 8·36-s − 37-s + 3·41-s + 2·43-s + 2·44-s + 3·47-s + 3·49-s − 10·52-s + 9·53-s − 6·59-s + 2·61-s + 4·63-s + 8·64-s − 11·67-s − 10·68-s + ⋯ |
| L(s) = 1 | − 4-s − 0.377·7-s − 4/3·9-s − 0.301·11-s + 1.38·13-s + 1.21·17-s − 0.688·19-s + 1.04·23-s − 2/5·25-s + 0.377·28-s − 0.928·29-s − 0.359·31-s + 4/3·36-s − 0.164·37-s + 0.468·41-s + 0.304·43-s + 0.301·44-s + 0.437·47-s + 3/7·49-s − 1.38·52-s + 1.23·53-s − 0.781·59-s + 0.256·61-s + 0.503·63-s + 64-s − 1.34·67-s − 1.21·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1109 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1109 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4409391126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4409391126\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 1109 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 24 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 19 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + p T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 37 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 52 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T + 55 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 61 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 74 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 19 T + 204 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 82 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 92 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
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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
−19.6965376313, −19.2220094270, −18.5364068137, −18.4051795439, −17.5000897239, −17.1788118360, −16.5008456052, −16.0320092906, −15.1275732218, −14.6546910416, −13.9935566066, −13.4553897361, −13.0478437957, −12.2771233676, −11.4521913371, −10.9019213354, −10.1688087125, −9.17350588568, −8.84376685905, −8.19035578738, −7.20394447877, −5.92321697996, −5.53873483112, −4.20481927453, −3.14611079748,
3.14611079748, 4.20481927453, 5.53873483112, 5.92321697996, 7.20394447877, 8.19035578738, 8.84376685905, 9.17350588568, 10.1688087125, 10.9019213354, 11.4521913371, 12.2771233676, 13.0478437957, 13.4553897361, 13.9935566066, 14.6546910416, 15.1275732218, 16.0320092906, 16.5008456052, 17.1788118360, 17.5000897239, 18.4051795439, 18.5364068137, 19.2220094270, 19.6965376313
