| L(s) = 1 | − 3·3-s + 6·9-s + 9·11-s + 17-s + 6·19-s + 9·25-s − 9·27-s − 27·33-s + 2·41-s − 12·43-s − 49-s − 3·51-s − 18·57-s − 21·59-s + 12·67-s + 18·73-s − 27·75-s + 9·81-s + 15·83-s + 17·89-s + 9·97-s + 54·99-s + 3·107-s − 25·113-s + 41·121-s − 6·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s + 2.71·11-s + 0.242·17-s + 1.37·19-s + 9/5·25-s − 1.73·27-s − 4.70·33-s + 0.312·41-s − 1.82·43-s − 1/7·49-s − 0.420·51-s − 2.38·57-s − 2.73·59-s + 1.46·67-s + 2.10·73-s − 3.11·75-s + 81-s + 1.64·83-s + 1.80·89-s + 0.913·97-s + 5.42·99-s + 0.290·107-s − 2.35·113-s + 3.72·121-s − 0.541·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.591008332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.591008332\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 137 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
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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
−7.994534862844667146489504830799, −7.67798577312615296581767990535, −6.99715914070005313004154236869, −6.60763922421494265864861705034, −6.51704221124947280465420006042, −6.13464432321114038245581166368, −5.38602791794981585011396149587, −5.04428668499925405063043158221, −4.70056033337817233269312321422, −4.06485135335111370186706839261, −3.55332217952919168892312602126, −3.12331478386812474268696405213, −1.87008030383107612363221050558, −1.22612190700549355602631101483, −0.816299219935570233708319377049,
0.816299219935570233708319377049, 1.22612190700549355602631101483, 1.87008030383107612363221050558, 3.12331478386812474268696405213, 3.55332217952919168892312602126, 4.06485135335111370186706839261, 4.70056033337817233269312321422, 5.04428668499925405063043158221, 5.38602791794981585011396149587, 6.13464432321114038245581166368, 6.51704221124947280465420006042, 6.60763922421494265864861705034, 6.99715914070005313004154236869, 7.67798577312615296581767990535, 7.994534862844667146489504830799
