| L(s) = 1 | + 4-s + 9-s − 2·13-s − 3·16-s + 2·25-s + 8·29-s + 36-s + 4·43-s + 10·49-s − 2·52-s + 12·53-s + 4·61-s − 7·64-s + 4·79-s + 81-s + 2·100-s − 24·101-s + 12·113-s + 8·116-s − 2·117-s + 121-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s − 0.554·13-s − 3/4·16-s + 2/5·25-s + 1.48·29-s + 1/6·36-s + 0.609·43-s + 10/7·49-s − 0.277·52-s + 1.64·53-s + 0.512·61-s − 7/8·64-s + 0.450·79-s + 1/9·81-s + 1/5·100-s − 2.38·101-s + 1.12·113-s + 0.742·116-s − 0.184·117-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/4·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.854640035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.854640035\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
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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
−9.034675924042442143095749364854, −8.751255841935656095454046788480, −8.201720666313246234236012947159, −7.64207526479637796430779583830, −7.06106563280908413208522152193, −6.89083570242035191053138253645, −6.32264419475329643065103807654, −5.66441761296988050389009552569, −5.19940310794129448841020654032, −4.43271259079955941847260762295, −4.19246927836238336661479687820, −3.23556710348062248878122127648, −2.58345108108264936945916302095, −2.06329049460713332419606765221, −0.899003034750625609668785182244,
0.899003034750625609668785182244, 2.06329049460713332419606765221, 2.58345108108264936945916302095, 3.23556710348062248878122127648, 4.19246927836238336661479687820, 4.43271259079955941847260762295, 5.19940310794129448841020654032, 5.66441761296988050389009552569, 6.32264419475329643065103807654, 6.89083570242035191053138253645, 7.06106563280908413208522152193, 7.64207526479637796430779583830, 8.201720666313246234236012947159, 8.751255841935656095454046788480, 9.034675924042442143095749364854
