L(s) = 1 | + 4-s − 9-s − 3·13-s + 16-s − 5·17-s + 4·19-s − 7·25-s − 36-s − 8·43-s − 12·47-s − 3·49-s − 3·52-s − 53-s − 12·59-s + 64-s − 16·67-s − 5·68-s + 4·76-s − 8·81-s + 28·83-s − 7·100-s − 2·101-s + 3·117-s − 10·121-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 1/3·9-s − 0.832·13-s + 1/4·16-s − 1.21·17-s + 0.917·19-s − 7/5·25-s − 1/6·36-s − 1.21·43-s − 1.75·47-s − 3/7·49-s − 0.416·52-s − 0.137·53-s − 1.56·59-s + 1/8·64-s − 1.95·67-s − 0.606·68-s + 0.458·76-s − 8/9·81-s + 3.07·83-s − 0.699·100-s − 0.199·101-s + 0.277·117-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 39 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
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
−9.024297880140288752033514422158, −8.200823256107863932218203065493, −7.943577710210549949158302907887, −7.49793203213281639607715926504, −6.85497752837072961087261959685, −6.53676710198420011340170421080, −5.94853182652613110701521894262, −5.44471179481293877004336907851, −4.76507370481753523622266990779, −4.43625652694183173663756275180, −3.44833700322933695312312050077, −3.07562043588201141617055326461, −2.21484940350145742309478095088, −1.63168325857678582994211492209, 0,
1.63168325857678582994211492209, 2.21484940350145742309478095088, 3.07562043588201141617055326461, 3.44833700322933695312312050077, 4.43625652694183173663756275180, 4.76507370481753523622266990779, 5.44471179481293877004336907851, 5.94853182652613110701521894262, 6.53676710198420011340170421080, 6.85497752837072961087261959685, 7.49793203213281639607715926504, 7.943577710210549949158302907887, 8.200823256107863932218203065493, 9.024297880140288752033514422158