L(s) = 1 | + 3-s − 2·5-s − 4·13-s − 2·15-s − 6·17-s − 4·19-s − 4·23-s + 5·25-s − 27-s + 12·29-s − 8·31-s + 10·37-s − 4·39-s − 20·41-s − 24·43-s − 8·47-s − 6·51-s − 6·53-s − 4·57-s + 4·59-s + 10·61-s + 8·65-s + 12·67-s − 4·69-s − 8·71-s − 2·73-s + 5·75-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.10·13-s − 0.516·15-s − 1.45·17-s − 0.917·19-s − 0.834·23-s + 25-s − 0.192·27-s + 2.22·29-s − 1.43·31-s + 1.64·37-s − 0.640·39-s − 3.12·41-s − 3.65·43-s − 1.16·47-s − 0.840·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s + 1.28·61-s + 0.992·65-s + 1.46·67-s − 0.481·69-s − 0.949·71-s − 0.234·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.693336054966773211165345744510, −8.452852375490863791022552046790, −8.004955919431920395231516056437, −7.925142282683276492331272005405, −7.21933587508136672693917626044, −6.67920969154841358656308281094, −6.56546049101523144679605423150, −6.52517704251871020369735813298, −5.46333067805441061726791650234, −5.09138610126330702360336062811, −4.62258012394729284856484925612, −4.57607406380260119161490479140, −3.74435846014854430131322311208, −3.54607926809291387447820047858, −2.96177544590213549371858414276, −2.44660670318111209718125024025, −2.00995147109164383869734732362, −1.39270698049319159371833539633, 0, 0,
1.39270698049319159371833539633, 2.00995147109164383869734732362, 2.44660670318111209718125024025, 2.96177544590213549371858414276, 3.54607926809291387447820047858, 3.74435846014854430131322311208, 4.57607406380260119161490479140, 4.62258012394729284856484925612, 5.09138610126330702360336062811, 5.46333067805441061726791650234, 6.52517704251871020369735813298, 6.56546049101523144679605423150, 6.67920969154841358656308281094, 7.21933587508136672693917626044, 7.925142282683276492331272005405, 8.004955919431920395231516056437, 8.452852375490863791022552046790, 8.693336054966773211165345744510