| L(s) = 1 | + 4-s − 3·9-s + 16·16-s + 96·19-s − 24·29-s + 78·31-s − 3·36-s + 49·49-s − 144·59-s + 252·61-s + 47·64-s + 36·71-s + 96·76-s + 154·79-s − 198·89-s + 576·101-s − 256·109-s − 24·116-s + 242·121-s + 78·124-s + 127-s + 131-s + 137-s + 139-s − 48·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/4·4-s − 1/3·9-s + 16-s + 5.05·19-s − 0.827·29-s + 2.51·31-s − 0.0833·36-s + 49-s − 2.44·59-s + 4.13·61-s + 0.734·64-s + 0.507·71-s + 1.26·76-s + 1.94·79-s − 2.22·89-s + 5.70·101-s − 2.34·109-s − 0.206·116-s + 2·121-s + 0.629·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1/3·144-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.531856270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.531856270\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 15 T^{4} - p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 326 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 335 T^{2} + 28704 T^{4} - 335 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 37 T + p^{2} T^{2} )^{2}( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 833 T^{2} + 414048 T^{4} + 833 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 39 T + 1468 T^{2} - 39 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 24 T - 793 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )( 1 + 24 T - 793 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 41 | $C_2^2$ | \( ( 1 - 2039 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 2542 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 3743 T^{2} + 9130368 T^{4} - 3743 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 3854 T^{2} + 6962835 T^{4} + 3854 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 72 T + 5209 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 126 T + 9013 T^{2} - 126 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 142 T^{2} - 20130957 T^{4} + 142 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p^{2} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 10610 T^{2} + 84173859 T^{4} - 10610 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 77 T - 312 T^{2} - 77 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 7390 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 99 T + 11188 T^{2} + 99 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 9071 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−7.72333100014800725364203670160, −7.47269715949661385100936572492, −7.17684881711044641838250872257, −7.00783194025909426909849541360, −6.58707564048309617788210944968, −6.44427036033926461941281063624, −6.20656388461547279360707030948, −5.76501443292481956653696198689, −5.53878357910941885970979964523, −5.48279359194409316047714355948, −5.16577600018367567833224361167, −4.96252082785889296498155795079, −4.76038098823084684984686315483, −4.31527718365009102169297420455, −3.77400915939293341599789398329, −3.68799067938388297300046206925, −3.41652916156734927411708066953, −3.14541816152167070867027225674, −2.83928672970195085903151881573, −2.50070096929673813897069556578, −2.26172300560981632117395174950, −1.48167969001001100221929291094, −1.23252833976567467564242824834, −0.845120233311817052126922017310, −0.63465356731674865545549840714,
0.63465356731674865545549840714, 0.845120233311817052126922017310, 1.23252833976567467564242824834, 1.48167969001001100221929291094, 2.26172300560981632117395174950, 2.50070096929673813897069556578, 2.83928672970195085903151881573, 3.14541816152167070867027225674, 3.41652916156734927411708066953, 3.68799067938388297300046206925, 3.77400915939293341599789398329, 4.31527718365009102169297420455, 4.76038098823084684984686315483, 4.96252082785889296498155795079, 5.16577600018367567833224361167, 5.48279359194409316047714355948, 5.53878357910941885970979964523, 5.76501443292481956653696198689, 6.20656388461547279360707030948, 6.44427036033926461941281063624, 6.58707564048309617788210944968, 7.00783194025909426909849541360, 7.17684881711044641838250872257, 7.47269715949661385100936572492, 7.72333100014800725364203670160
