| L(s) = 1 | + 6·7-s − 14·13-s − 76·19-s − 42·25-s + 20·31-s + 252·37-s + 100·43-s + 107·49-s − 158·61-s − 154·67-s − 68·73-s + 22·79-s − 84·91-s + 194·97-s − 266·103-s + 520·109-s + 150·121-s + 127-s + 131-s − 456·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 6/7·7-s − 1.07·13-s − 4·19-s − 1.67·25-s + 0.645·31-s + 6.81·37-s + 2.32·43-s + 2.18·49-s − 2.59·61-s − 2.29·67-s − 0.931·73-s + 0.278·79-s − 0.923·91-s + 2·97-s − 2.58·103-s + 4.77·109-s + 1.23·121-s + 0.00787·127-s + 0.00763·131-s − 3.42·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.260636724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.260636724\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 42 T^{2} + 1139 T^{4} + 42 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 3 T - 40 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 150 T^{2} + 7859 T^{4} - 150 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 7 T - 120 T^{2} + 7 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 378 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 1050 T^{2} + 822659 T^{4} + 1050 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 910 T^{2} + 120819 T^{4} - 910 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 10 T - 861 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 63 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 770 T^{2} - 2232861 T^{4} + 770 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T + 651 T^{2} - 50 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 2618 T^{2} + 1974243 T^{4} + 2618 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 210 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 2838 T^{2} - 4063117 T^{4} - 2838 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 79 T + 2520 T^{2} + 79 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 77 T + 1440 T^{2} + 77 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 3810 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )^{2}( 1 + 131 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 12210 T^{2} + 101625779 T^{4} + 12210 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 14042 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{4}( 1 + p T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.54735018751429241578265043701, −7.30211874874720807274759747980, −7.03555840131383225820419604932, −6.47387527085755567522738955974, −6.22675325283382149308762820101, −6.11893879132962074803506798465, −6.06047954547589638769314100909, −6.01296688035664149847174783529, −5.51772505009637172269894820641, −5.24400762863239872880130653261, −4.60355872574662389896616011506, −4.46904876366091035647250011598, −4.42726042639201521267573725102, −4.42154531834628450709900512274, −3.97301554363362746631238963667, −3.92365290992001381831183919651, −3.13354507713958997489743670752, −2.76217406571149991783098158862, −2.59809377083241434429203207382, −2.37858769347778194357885008348, −1.99432381617619005104056193467, −1.89728833558244654693023144443, −1.04722363394917587348122482322, −0.870306946084058631517255545813, −0.20780305641235744690208732172,
0.20780305641235744690208732172, 0.870306946084058631517255545813, 1.04722363394917587348122482322, 1.89728833558244654693023144443, 1.99432381617619005104056193467, 2.37858769347778194357885008348, 2.59809377083241434429203207382, 2.76217406571149991783098158862, 3.13354507713958997489743670752, 3.92365290992001381831183919651, 3.97301554363362746631238963667, 4.42154531834628450709900512274, 4.42726042639201521267573725102, 4.46904876366091035647250011598, 4.60355872574662389896616011506, 5.24400762863239872880130653261, 5.51772505009637172269894820641, 6.01296688035664149847174783529, 6.06047954547589638769314100909, 6.11893879132962074803506798465, 6.22675325283382149308762820101, 6.47387527085755567522738955974, 7.03555840131383225820419604932, 7.30211874874720807274759747980, 7.54735018751429241578265043701
