L(s) = 1 | − 2-s − 3-s − 4-s + 2·5-s + 6-s + 4·7-s + 3·8-s − 2·9-s − 2·10-s + 12-s − 4·14-s − 2·15-s − 16-s + 2·17-s + 2·18-s + 4·19-s − 2·20-s − 4·21-s − 3·24-s − 3·25-s + 2·27-s − 4·28-s + 2·30-s − 5·32-s − 2·34-s + 8·35-s + 2·36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s + 1.51·7-s + 1.06·8-s − 2/3·9-s − 0.632·10-s + 0.288·12-s − 1.06·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.471·18-s + 0.917·19-s − 0.447·20-s − 0.872·21-s − 0.612·24-s − 3/5·25-s + 0.384·27-s − 0.755·28-s + 0.365·30-s − 0.883·32-s − 0.342·34-s + 1.35·35-s + 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.193263731\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.193263731\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 53 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 103 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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
−8.553900202097024101892916437176, −8.048860814486580245989620714867, −7.58931156546967098088495922730, −7.43789373245577066589238154657, −6.65286287723150000318259691782, −5.99190753089947206603605134235, −5.55352926625868574311342302718, −5.41798539240380860950126931939, −4.72515894793674202931866028577, −4.44447133404808793949844223755, −3.63644220835827721308280057347, −2.92122792171412648845382441658, −1.98644552801823077734916819016, −1.57529317101823182442787469255, −0.70529753636413698670543663922,
0.70529753636413698670543663922, 1.57529317101823182442787469255, 1.98644552801823077734916819016, 2.92122792171412648845382441658, 3.63644220835827721308280057347, 4.44447133404808793949844223755, 4.72515894793674202931866028577, 5.41798539240380860950126931939, 5.55352926625868574311342302718, 5.99190753089947206603605134235, 6.65286287723150000318259691782, 7.43789373245577066589238154657, 7.58931156546967098088495922730, 8.048860814486580245989620714867, 8.553900202097024101892916437176