| L(s) = 1 | + 8·5-s − 5·9-s + 2·13-s + 6·17-s + 38·25-s + 10·29-s + 4·37-s − 16·41-s − 40·45-s − 5·49-s + 2·53-s − 4·61-s + 16·65-s + 18·73-s + 16·81-s + 48·85-s − 4·97-s − 4·101-s + 30·109-s + 28·113-s − 10·117-s − 18·121-s + 136·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 3.57·5-s − 5/3·9-s + 0.554·13-s + 1.45·17-s + 38/5·25-s + 1.85·29-s + 0.657·37-s − 2.49·41-s − 5.96·45-s − 5/7·49-s + 0.274·53-s − 0.512·61-s + 1.98·65-s + 2.10·73-s + 16/9·81-s + 5.20·85-s − 0.406·97-s − 0.398·101-s + 2.87·109-s + 2.63·113-s − 0.924·117-s − 1.63·121-s + 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.825279813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.825279813\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−8.264845552131759886230905508728, −7.41068992206199799917476516665, −6.67486851525401293685577568233, −6.41602681477419049736409749590, −6.17646264461499634926610391599, −5.70311644291868843897751049539, −5.50702009116960948045704530045, −4.89838836690724431377597203004, −4.86338510531971803230901631456, −3.35219753314436267860210658260, −3.27846026315236304587310905813, −2.45747388367035991238545338827, −2.31224961625843070084602752549, −1.49207601742473373563863301755, −1.03106032747078173831338248002,
1.03106032747078173831338248002, 1.49207601742473373563863301755, 2.31224961625843070084602752549, 2.45747388367035991238545338827, 3.27846026315236304587310905813, 3.35219753314436267860210658260, 4.86338510531971803230901631456, 4.89838836690724431377597203004, 5.50702009116960948045704530045, 5.70311644291868843897751049539, 6.17646264461499634926610391599, 6.41602681477419049736409749590, 6.67486851525401293685577568233, 7.41068992206199799917476516665, 8.264845552131759886230905508728
