L(s) = 1 | + 3-s − 7-s + 2·13-s − 19-s − 21-s − 25-s − 27-s + 31-s − 37-s + 2·39-s + 2·43-s − 57-s + 2·61-s − 67-s + 73-s − 75-s + 79-s − 81-s − 2·91-s + 93-s + 4·97-s + 103-s − 109-s − 111-s − 121-s + 127-s + 2·129-s + ⋯ |
L(s) = 1 | + 3-s − 7-s + 2·13-s − 19-s − 21-s − 25-s − 27-s + 31-s − 37-s + 2·39-s + 2·43-s − 57-s + 2·61-s − 67-s + 73-s − 75-s + 79-s − 81-s − 2·91-s + 93-s + 4·97-s + 103-s − 109-s − 111-s − 121-s + 127-s + 2·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.359520331\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.359520331\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$ | \( ( 1 - T )^{4} \) |
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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
−9.997954344333303524717599043405, −9.383467158233869330309434519483, −9.170360850273264025598008120593, −8.808915666656241617420891102977, −8.506554621931060771096713992486, −7.960070583648503875698738693212, −7.86080495463675631444327463042, −7.19652795889189431779836154334, −6.53153640417336782669873061070, −6.46804651908342530034393502210, −5.86860027522183108490775221435, −5.68546210133623193923663978823, −4.91505695290780064439623235451, −4.20038556684155557208127900518, −3.88051996524180013920209145277, −3.45768450465414803479217889066, −3.11227141670397092014620725843, −2.32619378445995974936277107524, −2.00816874465665383006964055389, −0.971873684487282232698546411294,
0.971873684487282232698546411294, 2.00816874465665383006964055389, 2.32619378445995974936277107524, 3.11227141670397092014620725843, 3.45768450465414803479217889066, 3.88051996524180013920209145277, 4.20038556684155557208127900518, 4.91505695290780064439623235451, 5.68546210133623193923663978823, 5.86860027522183108490775221435, 6.46804651908342530034393502210, 6.53153640417336782669873061070, 7.19652795889189431779836154334, 7.86080495463675631444327463042, 7.960070583648503875698738693212, 8.506554621931060771096713992486, 8.808915666656241617420891102977, 9.170360850273264025598008120593, 9.383467158233869330309434519483, 9.997954344333303524717599043405