L(s) = 1 | − 4-s + 4·7-s − 3·9-s + 16-s + 8·19-s − 25-s − 4·28-s + 4·31-s + 3·36-s − 2·49-s − 12·63-s − 64-s + 8·67-s − 8·76-s + 9·81-s + 4·97-s + 100-s − 4·103-s + 4·109-s + 4·112-s + 14·121-s − 4·124-s + 127-s + 131-s + 32·133-s + 137-s + 139-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 1.51·7-s − 9-s + 1/4·16-s + 1.83·19-s − 1/5·25-s − 0.755·28-s + 0.718·31-s + 1/2·36-s − 2/7·49-s − 1.51·63-s − 1/8·64-s + 0.977·67-s − 0.917·76-s + 81-s + 0.406·97-s + 1/10·100-s − 0.394·103-s + 0.383·109-s + 0.377·112-s + 1.27·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.025920571\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.025920571\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 31 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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
−8.166675189137703019004001484923, −7.903750616970958376074875267978, −7.49141713873053479915595282434, −6.95166010100213944744151041617, −6.36983599610703647960989892450, −5.80525900757485624726555403646, −5.38632415451667964893247277298, −5.05437751319379347313758860174, −4.64916903785667003185273060010, −4.09093799859713103651942115757, −3.35740457877933336906110113087, −3.00500477363704398491710942834, −2.20362344885623521787041062141, −1.50574124793191160132132115875, −0.72431480240733306960638962231,
0.72431480240733306960638962231, 1.50574124793191160132132115875, 2.20362344885623521787041062141, 3.00500477363704398491710942834, 3.35740457877933336906110113087, 4.09093799859713103651942115757, 4.64916903785667003185273060010, 5.05437751319379347313758860174, 5.38632415451667964893247277298, 5.80525900757485624726555403646, 6.36983599610703647960989892450, 6.95166010100213944744151041617, 7.49141713873053479915595282434, 7.903750616970958376074875267978, 8.166675189137703019004001484923