| L(s) = 1 | + 32·13-s − 10·25-s − 8·37-s − 74·49-s − 416·61-s + 244·73-s + 412·97-s + 152·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 36·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2.46·13-s − 2/5·25-s − 0.216·37-s − 1.51·49-s − 6.81·61-s + 3.34·73-s + 4.24·97-s + 1.39·109-s − 0.0165·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.213·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.390430370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.390430370\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + p T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 37 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 182 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 1502 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1787 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 1742 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 3158 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 3446 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 5213 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 3074 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 104 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 4118 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 6194 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 61 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 3842 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 7703 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 5938 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 103 T + p^{2} T^{2} )^{4} \) |
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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
−6.46714516588272109471462504129, −6.22168114863865587815448351150, −5.92553645230919867364688267683, −5.91013711555408519292102559470, −5.74050288954074986026344651373, −5.53694175776632457968551906662, −4.98203935184785359111739086473, −4.79338074044487663673143746573, −4.75274042403441174739100875855, −4.57168334969646327102594620875, −4.34808947721387432917751022335, −3.74049585135861123837115209434, −3.71574570850063050314310447410, −3.68489600185974284038891204691, −3.14181837639446801885348312499, −3.09699210620809449499052393467, −3.07591883702162120478574104304, −2.37215072993284779222685424203, −2.18939311894684469141543399841, −1.80996296196706377800845717985, −1.53201120816612171372504517416, −1.48150001799807314259330827012, −0.858315317033548896129516147632, −0.77462535524432316909202571365, −0.14526050983063020521515213326,
0.14526050983063020521515213326, 0.77462535524432316909202571365, 0.858315317033548896129516147632, 1.48150001799807314259330827012, 1.53201120816612171372504517416, 1.80996296196706377800845717985, 2.18939311894684469141543399841, 2.37215072993284779222685424203, 3.07591883702162120478574104304, 3.09699210620809449499052393467, 3.14181837639446801885348312499, 3.68489600185974284038891204691, 3.71574570850063050314310447410, 3.74049585135861123837115209434, 4.34808947721387432917751022335, 4.57168334969646327102594620875, 4.75274042403441174739100875855, 4.79338074044487663673143746573, 4.98203935184785359111739086473, 5.53694175776632457968551906662, 5.74050288954074986026344651373, 5.91013711555408519292102559470, 5.92553645230919867364688267683, 6.22168114863865587815448351150, 6.46714516588272109471462504129
