| L(s) = 1 | + 3·2-s + 5·4-s + 6·8-s − 9-s + 3·11-s − 2·13-s + 6·16-s − 3·18-s − 3·19-s + 9·22-s − 23-s − 25-s − 6·26-s + 6·32-s − 5·36-s − 9·38-s + 15·44-s − 3·46-s + 49-s − 3·50-s − 10·52-s + 7·64-s − 6·72-s − 15·76-s − 2·79-s + 18·88-s − 5·92-s + ⋯ |
| L(s) = 1 | + 3·2-s + 5·4-s + 6·8-s − 9-s + 3·11-s − 2·13-s + 6·16-s − 3·18-s − 3·19-s + 9·22-s − 23-s − 25-s − 6·26-s + 6·32-s − 5·36-s − 9·38-s + 15·44-s − 3·46-s + 49-s − 3·50-s − 10·52-s + 7·64-s − 6·72-s − 15·76-s − 2·79-s + 18·88-s − 5·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1054729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1054729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.805396083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.805396083\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| 79 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
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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
−10.55642939040921431880894596621, −10.00448774690162127733916602745, −9.690139282006946312412273682980, −9.107797467003673125038143348470, −8.512005525064109850489078268974, −8.389572118370019956343451187813, −7.45587824586107087242247156138, −7.14107941901366557069142286522, −6.55620991250325944382274391664, −6.35913235075795452626262381127, −5.83662928953082551647687125236, −5.83609080044802931961822375988, −4.92253435991809642512687876869, −4.49986360416101392454588295486, −4.29619829948425761964499908438, −3.79810701595247815614615475614, −3.57063253832746514693366163693, −2.66197523849166724621997452294, −2.27441734731518027309007583042, −1.75561980511686312460498537700,
1.75561980511686312460498537700, 2.27441734731518027309007583042, 2.66197523849166724621997452294, 3.57063253832746514693366163693, 3.79810701595247815614615475614, 4.29619829948425761964499908438, 4.49986360416101392454588295486, 4.92253435991809642512687876869, 5.83609080044802931961822375988, 5.83662928953082551647687125236, 6.35913235075795452626262381127, 6.55620991250325944382274391664, 7.14107941901366557069142286522, 7.45587824586107087242247156138, 8.389572118370019956343451187813, 8.512005525064109850489078268974, 9.107797467003673125038143348470, 9.690139282006946312412273682980, 10.00448774690162127733916602745, 10.55642939040921431880894596621
