| L(s) = 1 | − 6·5-s − 6·7-s − 3·11-s + 4·13-s + 19·25-s − 6·29-s + 36·35-s − 4·37-s + 9·41-s − 9·43-s + 12·47-s + 17·49-s + 18·55-s − 15·59-s + 8·61-s − 24·65-s + 15·67-s + 12·71-s − 22·73-s + 18·77-s + 6·79-s + 12·83-s − 24·91-s − 13·97-s + 18·101-s + 24·103-s + 6·107-s + ⋯ |
| L(s) = 1 | − 2.68·5-s − 2.26·7-s − 0.904·11-s + 1.10·13-s + 19/5·25-s − 1.11·29-s + 6.08·35-s − 0.657·37-s + 1.40·41-s − 1.37·43-s + 1.75·47-s + 17/7·49-s + 2.42·55-s − 1.95·59-s + 1.02·61-s − 2.97·65-s + 1.83·67-s + 1.42·71-s − 2.57·73-s + 2.05·77-s + 0.675·79-s + 1.31·83-s − 2.51·91-s − 1.31·97-s + 1.79·101-s + 2.36·103-s + 0.580·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2749313332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2749313332\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 9 T + 70 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 6 T + 91 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−9.253542531866835918691305774595, −9.225372151434558477169559033305, −8.655598750842081782487083748958, −8.247318681222401571380142768445, −7.989117999096146846008441345305, −7.39782548922027803675663333441, −7.28827038735117538891449799997, −6.91492001838096773895601695698, −6.26812505487645994004940357050, −6.07330681349118135528612935377, −5.53039100992771383728790366875, −4.85975193573797985228309119711, −4.42746028408117879293741535824, −3.80791135278475011820719256269, −3.53485522255024667314453387194, −3.45949075422610355847758834718, −2.86203641804073822991278420309, −2.19607955823744858583892575526, −0.888138492619801174951863627475, −0.26514203081634512701661242326,
0.26514203081634512701661242326, 0.888138492619801174951863627475, 2.19607955823744858583892575526, 2.86203641804073822991278420309, 3.45949075422610355847758834718, 3.53485522255024667314453387194, 3.80791135278475011820719256269, 4.42746028408117879293741535824, 4.85975193573797985228309119711, 5.53039100992771383728790366875, 6.07330681349118135528612935377, 6.26812505487645994004940357050, 6.91492001838096773895601695698, 7.28827038735117538891449799997, 7.39782548922027803675663333441, 7.989117999096146846008441345305, 8.247318681222401571380142768445, 8.655598750842081782487083748958, 9.225372151434558477169559033305, 9.253542531866835918691305774595
