| L(s) = 1 | − 4·3-s − 4-s + 4·5-s + 2·7-s + 8·9-s + 2·11-s + 4·12-s − 6·13-s − 16·15-s + 16-s + 8·17-s + 12·19-s − 4·20-s − 8·21-s + 8·23-s + 11·25-s − 12·27-s − 2·28-s + 10·31-s − 8·33-s + 8·35-s − 8·36-s − 4·37-s + 24·39-s − 8·41-s + 10·43-s − 2·44-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1/2·4-s + 1.78·5-s + 0.755·7-s + 8/3·9-s + 0.603·11-s + 1.15·12-s − 1.66·13-s − 4.13·15-s + 1/4·16-s + 1.94·17-s + 2.75·19-s − 0.894·20-s − 1.74·21-s + 1.66·23-s + 11/5·25-s − 2.30·27-s − 0.377·28-s + 1.79·31-s − 1.39·33-s + 1.35·35-s − 4/3·36-s − 0.657·37-s + 3.84·39-s − 1.24·41-s + 1.52·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.596737263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.596737263\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−10.30715306668908615105353651886, −10.00371010594340078478945396142, −9.469860622268719595884411198230, −9.377954911673261515456106229707, −8.999151651057373609621922748687, −7.988243755968983651350483261122, −7.53481493267558817117117696348, −7.37446780398770848515381582535, −6.66242122123468850683865267962, −6.29513176129576324935685193468, −5.66372414234957012284645163037, −5.64619258264652646641822030823, −5.00870411550505790812918215383, −4.87857414728819912306470687655, −4.68587487526895642164839448704, −3.20167947434558898002442533448, −3.11949885079916386525351164091, −1.92889400308110805666045385485, −1.09013030449873655382136290806, −0.943492535823683420003846303372,
0.943492535823683420003846303372, 1.09013030449873655382136290806, 1.92889400308110805666045385485, 3.11949885079916386525351164091, 3.20167947434558898002442533448, 4.68587487526895642164839448704, 4.87857414728819912306470687655, 5.00870411550505790812918215383, 5.64619258264652646641822030823, 5.66372414234957012284645163037, 6.29513176129576324935685193468, 6.66242122123468850683865267962, 7.37446780398770848515381582535, 7.53481493267558817117117696348, 7.988243755968983651350483261122, 8.999151651057373609621922748687, 9.377954911673261515456106229707, 9.469860622268719595884411198230, 10.00371010594340078478945396142, 10.30715306668908615105353651886
