| L(s) = 1 | + 4·5-s + 6·9-s + 11·25-s − 8·29-s + 20·41-s + 24·45-s + 14·49-s − 24·61-s + 27·81-s + 20·89-s − 40·101-s + 40·109-s − 22·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s − 32·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 2·9-s + 11/5·25-s − 1.48·29-s + 3.12·41-s + 3.57·45-s + 2·49-s − 3.07·61-s + 3·81-s + 2.11·89-s − 3.98·101-s + 3.83·109-s − 2·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.65·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.348249304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.348249304\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−9.637246109713695983875229439637, −9.560336945035571699201228338964, −9.124668931843461634877441046843, −9.076287668585436021240737243016, −8.229319891160462454634158521027, −7.64293236223942917262861386513, −7.34985575113537428293566201760, −7.14236779940067261021484832826, −6.39602811993377241350072145827, −6.22788357433274716568137446504, −5.64288232149954811348562845193, −5.45297476602907223592101191024, −4.61020434407118526590704215867, −4.46294500657258947843005443838, −3.89685840766965141864686436978, −3.24289412642377215021208331050, −2.42995270926065559315687496601, −2.13131395422938990165950266530, −1.46209794973462404933672720088, −0.980027824953639636672775212132,
0.980027824953639636672775212132, 1.46209794973462404933672720088, 2.13131395422938990165950266530, 2.42995270926065559315687496601, 3.24289412642377215021208331050, 3.89685840766965141864686436978, 4.46294500657258947843005443838, 4.61020434407118526590704215867, 5.45297476602907223592101191024, 5.64288232149954811348562845193, 6.22788357433274716568137446504, 6.39602811993377241350072145827, 7.14236779940067261021484832826, 7.34985575113537428293566201760, 7.64293236223942917262861386513, 8.229319891160462454634158521027, 9.076287668585436021240737243016, 9.124668931843461634877441046843, 9.560336945035571699201228338964, 9.637246109713695983875229439637
