L(s) = 1 | + 3-s − 2·4-s + 7-s − 2·12-s + 13-s + 3·16-s − 3·19-s + 21-s + 25-s − 27-s − 2·28-s + 39-s + 43-s + 3·48-s − 2·52-s − 3·57-s − 61-s − 4·64-s − 3·73-s + 75-s + 6·76-s + 2·79-s − 81-s − 2·84-s + 91-s − 3·97-s − 2·100-s + ⋯ |
L(s) = 1 | + 3-s − 2·4-s + 7-s − 2·12-s + 13-s + 3·16-s − 3·19-s + 21-s + 25-s − 27-s − 2·28-s + 39-s + 43-s + 3·48-s − 2·52-s − 3·57-s − 61-s − 4·64-s − 3·73-s + 75-s + 6·76-s + 2·79-s − 81-s − 2·84-s + 91-s − 3·97-s − 2·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5522146464\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5522146464\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 - T + T^{2} \) |
good | 2 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 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
−12.46799419406341853253916434073, −12.21013744596379579172905206964, −11.07613514157848440107508738172, −11.06044224702581829300215777349, −10.36961570628701526227411392803, −9.986493602966646755640094683693, −9.184746018849948894942346576850, −8.834935623988358616706225044344, −8.676571157915595226815636698647, −8.259617039379478357178920023381, −7.906844754017836480402007131656, −7.21151721168130198249337962751, −6.11878125073448870038652155030, −5.95423315817357622615728553006, −4.96317471180409279700529314242, −4.57745593534112865804498562797, −4.06639766111190915217367692288, −3.58034364638658197482065533815, −2.63246840570324538629877096217, −1.55532597752984363062862192735,
1.55532597752984363062862192735, 2.63246840570324538629877096217, 3.58034364638658197482065533815, 4.06639766111190915217367692288, 4.57745593534112865804498562797, 4.96317471180409279700529314242, 5.95423315817357622615728553006, 6.11878125073448870038652155030, 7.21151721168130198249337962751, 7.906844754017836480402007131656, 8.259617039379478357178920023381, 8.676571157915595226815636698647, 8.834935623988358616706225044344, 9.184746018849948894942346576850, 9.986493602966646755640094683693, 10.36961570628701526227411392803, 11.06044224702581829300215777349, 11.07613514157848440107508738172, 12.21013744596379579172905206964, 12.46799419406341853253916434073