| L(s) = 1 | − 2·3-s + 9-s + 6·11-s − 6·17-s − 8·19-s + 2·25-s + 4·27-s − 12·33-s + 8·41-s + 4·43-s + 10·49-s + 12·51-s + 16·57-s − 4·59-s + 4·67-s − 8·73-s − 4·75-s − 11·81-s + 12·83-s − 16·89-s + 12·97-s + 6·99-s − 2·107-s − 24·113-s + 6·121-s − 16·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 1.80·11-s − 1.45·17-s − 1.83·19-s + 2/5·25-s + 0.769·27-s − 2.08·33-s + 1.24·41-s + 0.609·43-s + 10/7·49-s + 1.68·51-s + 2.11·57-s − 0.520·59-s + 0.488·67-s − 0.936·73-s − 0.461·75-s − 1.22·81-s + 1.31·83-s − 1.69·89-s + 1.21·97-s + 0.603·99-s − 0.193·107-s − 2.25·113-s + 6/11·121-s − 1.44·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.19897703524912303983331836203, −6.78578991455553510747604347958, −6.54776865028719519718387340586, −6.13155366201468871723684955585, −5.95413535689012277680607297777, −5.32241022408675053405626678284, −4.76375644193032875713720822152, −4.33622280225177653425238433323, −4.09839914912718884687848426806, −3.65448852216624824299437944764, −2.68355806709530408668365008297, −2.33375611174102933798441977648, −1.56079065122177926114531174118, −0.888103737856071411479403727147, 0,
0.888103737856071411479403727147, 1.56079065122177926114531174118, 2.33375611174102933798441977648, 2.68355806709530408668365008297, 3.65448852216624824299437944764, 4.09839914912718884687848426806, 4.33622280225177653425238433323, 4.76375644193032875713720822152, 5.32241022408675053405626678284, 5.95413535689012277680607297777, 6.13155366201468871723684955585, 6.54776865028719519718387340586, 6.78578991455553510747604347958, 7.19897703524912303983331836203
