| L(s) = 1 | + 7-s − 9·23-s − 25-s − 8·31-s − 18·41-s − 9·47-s − 11·49-s − 9·71-s − 2·73-s + 79-s + 9·89-s + 10·97-s − 5·103-s + 18·113-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 9·161-s + 163-s + 167-s + 5·169-s + 173-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.87·23-s − 1/5·25-s − 1.43·31-s − 2.81·41-s − 1.31·47-s − 1.57·49-s − 1.06·71-s − 0.234·73-s + 0.112·79-s + 0.953·89-s + 1.01·97-s − 0.492·103-s + 1.69·113-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 0.709·161-s + 0.0783·163-s + 0.0773·167-s + 5/13·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.784587970178261364110271220861, −8.403666303646432396914750063462, −7.996430645039682495781148409352, −7.54126578340562370816930573688, −6.96041078722723265232200195035, −6.46072830779362298121358310868, −5.95053445968094770127731776879, −5.41322300491506834334438090389, −4.83853095850862908568367128713, −4.36624924232740593985812833037, −3.52157286255257416067222618936, −3.25761740385098948956313833972, −2.00834547823053848565276329050, −1.70864480387728740538837706740, 0,
1.70864480387728740538837706740, 2.00834547823053848565276329050, 3.25761740385098948956313833972, 3.52157286255257416067222618936, 4.36624924232740593985812833037, 4.83853095850862908568367128713, 5.41322300491506834334438090389, 5.95053445968094770127731776879, 6.46072830779362298121358310868, 6.96041078722723265232200195035, 7.54126578340562370816930573688, 7.996430645039682495781148409352, 8.403666303646432396914750063462, 8.784587970178261364110271220861
