| L(s) = 1 | − 2·4-s − 4·9-s − 11·13-s + 4·16-s + 2·17-s − 9·25-s + 6·29-s + 8·36-s − 8·37-s + 3·41-s + 6·49-s + 22·52-s + 10·53-s − 2·61-s − 8·64-s − 4·68-s − 8·73-s + 7·81-s − 23·89-s − 10·97-s + 18·100-s − 3·101-s − 3·109-s + 8·113-s − 12·116-s + 44·117-s + 5·121-s + ⋯ |
| L(s) = 1 | − 4-s − 4/3·9-s − 3.05·13-s + 16-s + 0.485·17-s − 9/5·25-s + 1.11·29-s + 4/3·36-s − 1.31·37-s + 0.468·41-s + 6/7·49-s + 3.05·52-s + 1.37·53-s − 0.256·61-s − 64-s − 0.485·68-s − 0.936·73-s + 7/9·81-s − 2.43·89-s − 1.01·97-s + 9/5·100-s − 0.298·101-s − 0.287·109-s + 0.752·113-s − 1.11·116-s + 4.06·117-s + 5/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + p T^{2} \) |
| 1049 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 15 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−10.55113662016391325739445842509, −10.05642914213417151082544374091, −9.726879411927656792494213943941, −9.208203683676293367164105847628, −8.526949565307739401075433286569, −8.077255579496139367077576728667, −7.43184936793255618520506019938, −6.96483861012459033328029358476, −5.73234972110240541498848654609, −5.53012406426133137540514554953, −4.80217468152932955759112686736, −4.15843180095822047437681722474, −3.08417728375381141496450972947, −2.34856180415169019684910670958, 0,
2.34856180415169019684910670958, 3.08417728375381141496450972947, 4.15843180095822047437681722474, 4.80217468152932955759112686736, 5.53012406426133137540514554953, 5.73234972110240541498848654609, 6.96483861012459033328029358476, 7.43184936793255618520506019938, 8.077255579496139367077576728667, 8.526949565307739401075433286569, 9.208203683676293367164105847628, 9.726879411927656792494213943941, 10.05642914213417151082544374091, 10.55113662016391325739445842509
