| L(s) = 1 | + 4-s − 2·13-s − 3·16-s + 25-s − 8·31-s + 4·37-s − 10·49-s − 2·52-s − 12·61-s − 7·64-s − 8·67-s − 4·73-s − 16·79-s − 4·97-s + 100-s − 8·103-s + 12·109-s + 10·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 4·148-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.554·13-s − 3/4·16-s + 1/5·25-s − 1.43·31-s + 0.657·37-s − 1.42·49-s − 0.277·52-s − 1.53·61-s − 7/8·64-s − 0.977·67-s − 0.468·73-s − 1.80·79-s − 0.406·97-s + 1/10·100-s − 0.788·103-s + 1.14·109-s + 0.909·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.328·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−8.586167524273127109025637919098, −7.964739217835558664032960662696, −7.46728862926630623201493867144, −7.22532339209785524309518881889, −6.67067715187431506355624848497, −6.18442823511588326637929893197, −5.73056471146513146878035825057, −5.13670257649801654424344526907, −4.55328363164642304319086054310, −4.18471191300724743757003683414, −3.29040309023981160399048120423, −2.87017096304548857061977088092, −2.11005368985475829882968048669, −1.48034033501363092040574115276, 0,
1.48034033501363092040574115276, 2.11005368985475829882968048669, 2.87017096304548857061977088092, 3.29040309023981160399048120423, 4.18471191300724743757003683414, 4.55328363164642304319086054310, 5.13670257649801654424344526907, 5.73056471146513146878035825057, 6.18442823511588326637929893197, 6.67067715187431506355624848497, 7.22532339209785524309518881889, 7.46728862926630623201493867144, 7.964739217835558664032960662696, 8.586167524273127109025637919098
