| L(s) = 1 | − 2·4-s − 3·7-s − 13-s − 6·19-s + 25-s + 6·28-s + 16·31-s + 4·37-s + 9·43-s − 49-s + 2·52-s + 3·61-s + 8·64-s − 67-s + 8·73-s + 12·76-s − 6·79-s + 3·91-s − 21·97-s − 2·100-s − 10·103-s + 5·121-s − 32·124-s + 127-s + 131-s + 18·133-s + 137-s + ⋯ |
| L(s) = 1 | − 4-s − 1.13·7-s − 0.277·13-s − 1.37·19-s + 1/5·25-s + 1.13·28-s + 2.87·31-s + 0.657·37-s + 1.37·43-s − 1/7·49-s + 0.277·52-s + 0.384·61-s + 64-s − 0.122·67-s + 0.936·73-s + 1.37·76-s − 0.675·79-s + 0.314·91-s − 2.13·97-s − 1/5·100-s − 0.985·103-s + 5/11·121-s − 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 1.56·133-s + 0.0854·137-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 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 121 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−8.480901082081669369313021234185, −8.222558174419868697956726042484, −7.73357726374294179786249343765, −6.94006346582863040803434307408, −6.62087108357089711931420485817, −6.22937698099774888739284160773, −5.71575621861714357614145644547, −5.02280132729883111843488321964, −4.43512239506149368647644719934, −4.24704295223949328583847853179, −3.55272654159207941978845654536, −2.74241854004648047767538747088, −2.40700196226583927677503229614, −1.03100249179179784668328901679, 0,
1.03100249179179784668328901679, 2.40700196226583927677503229614, 2.74241854004648047767538747088, 3.55272654159207941978845654536, 4.24704295223949328583847853179, 4.43512239506149368647644719934, 5.02280132729883111843488321964, 5.71575621861714357614145644547, 6.22937698099774888739284160773, 6.62087108357089711931420485817, 6.94006346582863040803434307408, 7.73357726374294179786249343765, 8.222558174419868697956726042484, 8.480901082081669369313021234185
