| L(s) = 1 | + 3-s − 2·4-s + 2·7-s + 9-s − 2·12-s − 9·19-s + 2·21-s − 2·25-s + 27-s − 4·28-s + 8·31-s − 2·36-s + 6·37-s − 17·43-s − 11·49-s − 9·57-s + 5·61-s + 2·63-s + 8·64-s + 3·67-s + 15·73-s − 2·75-s + 18·76-s + 6·79-s + 81-s − 4·84-s + 8·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.755·7-s + 1/3·9-s − 0.577·12-s − 2.06·19-s + 0.436·21-s − 2/5·25-s + 0.192·27-s − 0.755·28-s + 1.43·31-s − 1/3·36-s + 0.986·37-s − 2.59·43-s − 1.57·49-s − 1.19·57-s + 0.640·61-s + 0.251·63-s + 64-s + 0.366·67-s + 1.75·73-s − 0.230·75-s + 2.06·76-s + 0.675·79-s + 1/9·81-s − 0.436·84-s + 0.829·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3753 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3753 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7848168382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7848168382\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 139 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 152 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.79898066151345616563362413459, −12.03551430883680223051385888511, −11.34288910877518600164549702966, −10.81504134804583099608128460624, −9.897633572346004763616176246564, −9.678645628683394499327685422800, −8.651995222917061986286296695884, −8.384266449291099319923889655855, −7.950414705036598094436284171496, −6.80877420615012413665210872263, −6.19680654415688338201421798137, −4.88027168976124803250423883067, −4.58018872516037333394026897323, −3.60596105590450069614733606265, −2.14287191734425882650377924742,
2.14287191734425882650377924742, 3.60596105590450069614733606265, 4.58018872516037333394026897323, 4.88027168976124803250423883067, 6.19680654415688338201421798137, 6.80877420615012413665210872263, 7.950414705036598094436284171496, 8.384266449291099319923889655855, 8.651995222917061986286296695884, 9.678645628683394499327685422800, 9.897633572346004763616176246564, 10.81504134804583099608128460624, 11.34288910877518600164549702966, 12.03551430883680223051385888511, 12.79898066151345616563362413459
