| L(s) = 1 | + 3-s + 3·4-s − 2·9-s + 3·12-s + 3·13-s + 5·16-s + 4·17-s + 2·23-s − 25-s − 5·27-s − 5·29-s − 6·36-s + 3·39-s + 43-s + 5·48-s + 10·49-s + 4·51-s + 9·52-s − 3·53-s + 2·61-s + 3·64-s + 12·68-s + 2·69-s − 75-s + 20·79-s + 81-s − 5·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 3/2·4-s − 2/3·9-s + 0.866·12-s + 0.832·13-s + 5/4·16-s + 0.970·17-s + 0.417·23-s − 1/5·25-s − 0.962·27-s − 0.928·29-s − 36-s + 0.480·39-s + 0.152·43-s + 0.721·48-s + 10/7·49-s + 0.560·51-s + 1.24·52-s − 0.412·53-s + 0.256·61-s + 3/8·64-s + 1.45·68-s + 0.240·69-s − 0.115·75-s + 2.25·79-s + 1/9·81-s − 0.536·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.743668393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.743668393\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
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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
−9.283801181642202410413591162452, −9.118156213468938920515776705186, −8.323891983826709942529468421990, −8.036737260298134988632183430215, −7.44672290668665189495592445106, −7.14381309780439507288416349182, −6.41256036705977864383024029269, −6.00487561069713688977256469955, −5.57856709652993787067748081937, −4.91858184841305536432370860570, −3.73412072754412682263600197624, −3.54616616765813855891250107614, −2.70260133762913404142662562259, −2.21838709031619195362664474946, −1.28050226101779786725329812356,
1.28050226101779786725329812356, 2.21838709031619195362664474946, 2.70260133762913404142662562259, 3.54616616765813855891250107614, 3.73412072754412682263600197624, 4.91858184841305536432370860570, 5.57856709652993787067748081937, 6.00487561069713688977256469955, 6.41256036705977864383024029269, 7.14381309780439507288416349182, 7.44672290668665189495592445106, 8.036737260298134988632183430215, 8.323891983826709942529468421990, 9.118156213468938920515776705186, 9.283801181642202410413591162452
