L(s) = 1 | − 2·2-s + 2·4-s + 4·5-s − 4·8-s − 8·10-s − 4·13-s + 8·16-s + 14·17-s + 8·20-s + 5·25-s + 8·26-s + 30·29-s − 8·32-s − 28·34-s − 8·37-s − 16·40-s + 24·41-s − 10·50-s − 8·52-s − 10·53-s − 60·58-s − 12·61-s + 8·64-s − 16·65-s + 28·68-s − 22·73-s + 16·74-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s + 1.78·5-s − 1.41·8-s − 2.52·10-s − 1.10·13-s + 2·16-s + 3.39·17-s + 1.78·20-s + 25-s + 1.56·26-s + 5.57·29-s − 1.41·32-s − 4.80·34-s − 1.31·37-s − 2.52·40-s + 3.74·41-s − 1.41·50-s − 1.10·52-s − 1.37·53-s − 7.87·58-s − 1.53·61-s + 64-s − 1.98·65-s + 3.39·68-s − 2.57·73-s + 1.85·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.300859861\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.300859861\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
good | 3 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + 12 T + p T^{2} )^{2}( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} )( 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
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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
−8.772771927299445725384838735213, −8.371000029517526354214094418073, −8.166095409120280519768180337559, −8.131400654565639397295644617651, −8.031606914787126180474967362253, −7.30584786406429692685882214829, −7.27866651381943936682065003800, −7.13353433799032811832697317839, −6.44201946370872960518714546572, −6.35180832497297740861400420421, −6.12740126535897525430798517549, −5.96617526425806339880635493458, −5.50659100733429146511757047446, −5.25933939926732076545641023941, −5.17643850111345254423070282768, −4.68089880556546600524935083596, −4.12372830199504256312742633006, −4.07024148128909104046173651961, −3.10735196321887568127731194372, −2.78129623757159189916483365992, −2.77270746314136231897448015067, −2.74673051334977702377817061566, −1.55798000021671217553507804455, −1.35343866750416102201107010287, −0.866992351343229496059516783790,
0.866992351343229496059516783790, 1.35343866750416102201107010287, 1.55798000021671217553507804455, 2.74673051334977702377817061566, 2.77270746314136231897448015067, 2.78129623757159189916483365992, 3.10735196321887568127731194372, 4.07024148128909104046173651961, 4.12372830199504256312742633006, 4.68089880556546600524935083596, 5.17643850111345254423070282768, 5.25933939926732076545641023941, 5.50659100733429146511757047446, 5.96617526425806339880635493458, 6.12740126535897525430798517549, 6.35180832497297740861400420421, 6.44201946370872960518714546572, 7.13353433799032811832697317839, 7.27866651381943936682065003800, 7.30584786406429692685882214829, 8.031606914787126180474967362253, 8.131400654565639397295644617651, 8.166095409120280519768180337559, 8.371000029517526354214094418073, 8.772771927299445725384838735213