| L(s) = 1 | − 16·7-s + 2·9-s − 12·13-s − 16-s + 16·19-s − 12·31-s − 24·37-s + 12·43-s + 132·49-s − 24·61-s − 32·63-s + 20·79-s − 5·81-s + 192·91-s + 24·97-s + 56·109-s + 16·112-s − 24·117-s − 40·121-s + 127-s + 131-s − 256·133-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 6.04·7-s + 2/3·9-s − 3.32·13-s − 1/4·16-s + 3.67·19-s − 2.15·31-s − 3.94·37-s + 1.82·43-s + 18.8·49-s − 3.07·61-s − 4.03·63-s + 2.25·79-s − 5/9·81-s + 20.1·91-s + 2.43·97-s + 5.36·109-s + 1.51·112-s − 2.21·117-s − 3.63·121-s + 0.0887·127-s + 0.0873·131-s − 22.1·133-s + 0.0854·137-s + 0.0848·139-s − 1/6·144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2838920622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2838920622\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )( 1 + 16 T^{2} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 706 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )( 1 + 40 T^{2} + p^{2} T^{4} ) \) |
| 31 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 6286 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 14434 T^{4} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.01775091978974021704800066602, −6.97239273024662659244090820555, −6.66638591006437391324509327225, −6.49032805634132394371860172616, −6.08321833062587085739504964444, −5.97920358744723080760732481721, −5.75289344278902563946886756902, −5.72343930962737497177682735936, −5.18719675634285306238155786978, −4.93523821512521970069000322291, −4.88962321417188530006346456083, −4.73010517988797683512350664556, −3.92696382048441280782189319442, −3.80994254232110065551710715070, −3.63805441140259002068395332658, −3.32515553684443634556640942630, −3.23517271497079398645015930388, −3.00504829561743012591582822041, −2.95417104135067089236185922589, −2.50465419474535202668698768109, −2.12677550590039554762437282010, −1.89721627541678866850102460989, −1.07598828802837731205861724513, −0.38473371046296143372751368507, −0.29347935697828269337349601110,
0.29347935697828269337349601110, 0.38473371046296143372751368507, 1.07598828802837731205861724513, 1.89721627541678866850102460989, 2.12677550590039554762437282010, 2.50465419474535202668698768109, 2.95417104135067089236185922589, 3.00504829561743012591582822041, 3.23517271497079398645015930388, 3.32515553684443634556640942630, 3.63805441140259002068395332658, 3.80994254232110065551710715070, 3.92696382048441280782189319442, 4.73010517988797683512350664556, 4.88962321417188530006346456083, 4.93523821512521970069000322291, 5.18719675634285306238155786978, 5.72343930962737497177682735936, 5.75289344278902563946886756902, 5.97920358744723080760732481721, 6.08321833062587085739504964444, 6.49032805634132394371860172616, 6.66638591006437391324509327225, 6.97239273024662659244090820555, 7.01775091978974021704800066602
