| L(s) = 1 | − 4-s − 2·5-s − 2·7-s − 2·9-s + 3·11-s + 3·13-s − 3·16-s + 2·17-s − 3·19-s + 2·20-s + 6·23-s + 2·25-s + 2·28-s + 2·31-s + 4·35-s + 2·36-s − 9·37-s − 4·41-s − 5·43-s − 3·44-s + 4·45-s + 5·47-s − 2·49-s − 3·52-s + 8·53-s − 6·55-s − 6·61-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s − 0.755·7-s − 2/3·9-s + 0.904·11-s + 0.832·13-s − 3/4·16-s + 0.485·17-s − 0.688·19-s + 0.447·20-s + 1.25·23-s + 2/5·25-s + 0.377·28-s + 0.359·31-s + 0.676·35-s + 1/3·36-s − 1.47·37-s − 0.624·41-s − 0.762·43-s − 0.452·44-s + 0.596·45-s + 0.729·47-s − 2/7·49-s − 0.416·52-s + 1.09·53-s − 0.809·55-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1055 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1055 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4327118447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4327118447\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| 211 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_4$ | \( 1 + 5 T + 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T - 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 126 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 134 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−19.9148362304, −19.3107159065, −18.8402003524, −18.4817978283, −17.6009661112, −17.0156151653, −16.6569636285, −15.9034847717, −15.3841649967, −14.8560125278, −14.0342638187, −13.6280468723, −12.8820003840, −12.2461967163, −11.5872594975, −11.0580817106, −10.2565800442, −9.29172579638, −8.76322600786, −8.24857044116, −7.01058027544, −6.49283308991, −5.31273719521, −4.14306020986, −3.28760233096,
3.28760233096, 4.14306020986, 5.31273719521, 6.49283308991, 7.01058027544, 8.24857044116, 8.76322600786, 9.29172579638, 10.2565800442, 11.0580817106, 11.5872594975, 12.2461967163, 12.8820003840, 13.6280468723, 14.0342638187, 14.8560125278, 15.3841649967, 15.9034847717, 16.6569636285, 17.0156151653, 17.6009661112, 18.4817978283, 18.8402003524, 19.3107159065, 19.9148362304
