| L(s) = 1 | + 4·2-s + 4·3-s + 10·4-s + 16·6-s − 4·7-s + 20·8-s + 8·9-s − 12·11-s + 40·12-s − 16·14-s + 35·16-s + 32·18-s − 16·21-s − 48·22-s + 80·24-s − 8·25-s + 16·27-s − 40·28-s + 4·31-s + 56·32-s − 48·33-s + 80·36-s − 4·37-s + 12·41-s − 64·42-s − 32·43-s − 120·44-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2.30·3-s + 5·4-s + 6.53·6-s − 1.51·7-s + 7.07·8-s + 8/3·9-s − 3.61·11-s + 11.5·12-s − 4.27·14-s + 35/4·16-s + 7.54·18-s − 3.49·21-s − 10.2·22-s + 16.3·24-s − 8/5·25-s + 3.07·27-s − 7.55·28-s + 0.718·31-s + 9.89·32-s − 8.35·33-s + 40/3·36-s − 0.657·37-s + 1.87·41-s − 9.87·42-s − 4.87·43-s − 18.0·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.71557614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.71557614\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 16 T^{3} + 31 T^{4} - 16 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + 6 T + p T^{2} )^{2}( 1 + 14 T^{2} + p^{2} T^{4} ) \) |
| 13 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 22 T^{2} + 195 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 1567 T^{4} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 192 T^{3} - 1633 T^{4} + 192 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 12 T^{3} - 1138 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 660 T^{3} + 5854 T^{4} - 660 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 16 T + 132 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 166 T^{2} + 11235 T^{4} - 166 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 - 22 T^{2} + 3555 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1344 T^{3} + 13583 T^{4} - 1344 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 52 T^{2} - 714 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 1056 T^{3} + 15199 T^{4} - 1056 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - p^{2} T^{4} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1200 T^{3} + 11234 T^{4} - 1200 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 148 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 6 T + 115 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 768 T^{3} + 18431 T^{4} - 768 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−9.752014345218868879940925867867, −8.985078578134299110699811681977, −8.649428558016995130603437052587, −8.316137559443037282396789945132, −8.136173077134635711114609627570, −8.110515223557513335272537279947, −7.63140727303889405310702527362, −7.31376846637755401240778563737, −7.24881208561829318579061624362, −6.77301554738861111620207073434, −6.46646141370193084872077424014, −6.16160004400173022416582634342, −5.99556004703932204636632530834, −5.31411774197074858819220724737, −5.11121909804754919109036335268, −4.96644091721089275486421823469, −4.90478781771976075364589063787, −3.90277785743317142631101402068, −3.78158221610359380867086243472, −3.63773934402899945170112894190, −3.20445305336769339139509173356, −2.71206547130320650699993477891, −2.62454139074629107374236963162, −2.40197740993504597470768251406, −1.94856268676180420946848947263,
1.94856268676180420946848947263, 2.40197740993504597470768251406, 2.62454139074629107374236963162, 2.71206547130320650699993477891, 3.20445305336769339139509173356, 3.63773934402899945170112894190, 3.78158221610359380867086243472, 3.90277785743317142631101402068, 4.90478781771976075364589063787, 4.96644091721089275486421823469, 5.11121909804754919109036335268, 5.31411774197074858819220724737, 5.99556004703932204636632530834, 6.16160004400173022416582634342, 6.46646141370193084872077424014, 6.77301554738861111620207073434, 7.24881208561829318579061624362, 7.31376846637755401240778563737, 7.63140727303889405310702527362, 8.110515223557513335272537279947, 8.136173077134635711114609627570, 8.316137559443037282396789945132, 8.649428558016995130603437052587, 8.985078578134299110699811681977, 9.752014345218868879940925867867
