L(s) = 1 | + 4·7-s − 3·9-s + 3·13-s + 3·19-s + 2·25-s + 4·37-s − 8·43-s + 2·49-s + 12·61-s − 12·63-s + 8·67-s + 16·73-s + 16·79-s + 9·81-s + 12·91-s − 4·97-s − 8·103-s − 28·109-s − 9·117-s − 2·121-s + 127-s + 131-s + 12·133-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 1.51·7-s − 9-s + 0.832·13-s + 0.688·19-s + 2/5·25-s + 0.657·37-s − 1.21·43-s + 2/7·49-s + 1.53·61-s − 1.51·63-s + 0.977·67-s + 1.87·73-s + 1.80·79-s + 81-s + 1.25·91-s − 0.406·97-s − 0.788·103-s − 2.68·109-s − 0.832·117-s − 0.181·121-s + 0.0887·127-s + 0.0873·131-s + 1.04·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.898380263\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.898380263\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−9.377794860976983045047426414760, −8.677127898806730735574472290390, −8.246921843896952000873708795707, −8.066078596506952922567643310443, −7.58641323981143431134019551890, −6.64995135092008841393239799737, −6.53443716301468912394905016843, −5.61149766454949985540641147620, −5.23858374985387961733756037463, −4.90420053169359152825971812036, −4.04461079927145230319373061300, −3.50554236264963446858286436107, −2.71121107504986004347014841807, −1.93353644640575792468477108158, −1.01962578993384778489082765439,
1.01962578993384778489082765439, 1.93353644640575792468477108158, 2.71121107504986004347014841807, 3.50554236264963446858286436107, 4.04461079927145230319373061300, 4.90420053169359152825971812036, 5.23858374985387961733756037463, 5.61149766454949985540641147620, 6.53443716301468912394905016843, 6.64995135092008841393239799737, 7.58641323981143431134019551890, 8.066078596506952922567643310443, 8.246921843896952000873708795707, 8.677127898806730735574472290390, 9.377794860976983045047426414760