| L(s) = 1 | + 2·5-s + 10·7-s + 10·11-s − 101·25-s − 100·29-s − 6·31-s + 20·35-s − 109·49-s + 2·53-s + 20·55-s − 20·59-s + 130·73-s + 100·77-s + 76·79-s − 38·83-s − 70·97-s + 290·101-s + 140·103-s − 206·107-s − 441·121-s − 302·125-s + 127-s + 131-s + 137-s + 139-s − 200·145-s + 149-s + ⋯ |
| L(s) = 1 | + 2/5·5-s + 10/7·7-s + 0.909·11-s − 4.03·25-s − 3.44·29-s − 0.193·31-s + 4/7·35-s − 2.22·49-s + 2/53·53-s + 4/11·55-s − 0.338·59-s + 1.78·73-s + 1.29·77-s + 0.962·79-s − 0.457·83-s − 0.721·97-s + 2.87·101-s + 1.35·103-s − 1.92·107-s − 3.64·121-s − 2.41·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1.37·145-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.971515009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.971515009\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - T + 52 T^{2} - 3 T^{3} + 52 p^{2} T^{4} - p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 7 | \( ( 1 - 5 T + 92 T^{2} - 575 T^{3} + 92 p^{2} T^{4} - 5 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 11 | \( ( 1 - 5 T + 258 T^{2} - 1165 T^{3} + 258 p^{2} T^{4} - 5 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 13 | \( 1 - 334 T^{2} + 84335 T^{4} - 13780900 T^{6} + 84335 p^{4} T^{8} - 334 p^{8} T^{10} + p^{12} T^{12} \) |
| 17 | \( 1 - 398 T^{2} + 101631 T^{4} - 23365732 T^{6} + 101631 p^{4} T^{8} - 398 p^{8} T^{10} + p^{12} T^{12} \) |
| 19 | \( 1 - 862 T^{2} + 443951 T^{4} - 159427588 T^{6} + 443951 p^{4} T^{8} - 862 p^{8} T^{10} + p^{12} T^{12} \) |
| 23 | \( 1 - 1118 T^{2} + 1115871 T^{4} - 26343644 p T^{6} + 1115871 p^{4} T^{8} - 1118 p^{8} T^{10} + p^{12} T^{12} \) |
| 29 | \( ( 1 + 50 T + 3243 T^{2} + 86980 T^{3} + 3243 p^{2} T^{4} + 50 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 31 | \( ( 1 + 3 T + 1596 T^{2} + 21737 T^{3} + 1596 p^{2} T^{4} + 3 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 37 | \( 1 - 2254 T^{2} + 4355855 T^{4} - 6892125220 T^{6} + 4355855 p^{4} T^{8} - 2254 p^{8} T^{10} + p^{12} T^{12} \) |
| 41 | \( 1 - 3086 T^{2} + 944415 T^{4} + 5295185180 T^{6} + 944415 p^{4} T^{8} - 3086 p^{8} T^{10} + p^{12} T^{12} \) |
| 43 | \( 1 - 5014 T^{2} + 15555935 T^{4} - 35545688500 T^{6} + 15555935 p^{4} T^{8} - 5014 p^{8} T^{10} + p^{12} T^{12} \) |
| 47 | \( 1 - 6054 T^{2} + 22536015 T^{4} - 55171604180 T^{6} + 22536015 p^{4} T^{8} - 6054 p^{8} T^{10} + p^{12} T^{12} \) |
| 53 | \( ( 1 - T + 6444 T^{2} - 38171 T^{3} + 6444 p^{2} T^{4} - p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 59 | \( ( 1 + 10 T + 2983 T^{2} + 259980 T^{3} + 2983 p^{2} T^{4} + 10 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 61 | \( 1 - 1302 T^{2} + 12899391 T^{4} - 73977235508 T^{6} + 12899391 p^{4} T^{8} - 1302 p^{8} T^{10} + p^{12} T^{12} \) |
| 67 | \( 1 - 6774 T^{2} + 41880255 T^{4} - 256439253620 T^{6} + 41880255 p^{4} T^{8} - 6774 p^{8} T^{10} + p^{12} T^{12} \) |
| 71 | \( 1 - 24126 T^{2} + 267135135 T^{4} - 1719429450500 T^{6} + 267135135 p^{4} T^{8} - 24126 p^{8} T^{10} + p^{12} T^{12} \) |
| 73 | \( ( 1 - 65 T + 2562 T^{2} + 290255 T^{3} + 2562 p^{2} T^{4} - 65 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 79 | \( ( 1 - 38 T + 10211 T^{2} - 150572 T^{3} + 10211 p^{2} T^{4} - 38 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 83 | \( ( 1 + 19 T + 9634 T^{2} - 116901 T^{3} + 9634 p^{2} T^{4} + 19 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 89 | \( 1 - 34886 T^{2} + 591188655 T^{4} - 5920111746580 T^{6} + 591188655 p^{4} T^{8} - 34886 p^{8} T^{10} + p^{12} T^{12} \) |
| 97 | \( ( 1 + 35 T + 902 T^{2} + 852455 T^{3} + 902 p^{2} T^{4} + 35 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.18225539372939554376529407600, −5.12831259798362014630988071909, −4.88516677014278220127961736965, −4.66636934234438052566677196502, −4.66193978374645556770654987963, −4.14102312252380929625495713992, −3.97516071198196436787025068960, −3.96050267331164881936393987037, −3.95344568614688764481435283420, −3.79189537710122309498654629439, −3.64271520530884667706869018013, −3.21576370907211963372992069231, −3.01896079649014215695816029054, −2.93237805403089410064888819028, −2.78063666685823335185512965452, −2.12306927751757402615278975838, −2.06970862241595621924164299179, −1.97769646067554746484348826782, −1.90708784257719209813038225991, −1.63948706030288368478846581827, −1.58337647117500837594788947961, −1.13316767003147058429016751628, −0.868509716819243490267274627232, −0.33644015989172098564050735859, −0.19425794613896566575607310595,
0.19425794613896566575607310595, 0.33644015989172098564050735859, 0.868509716819243490267274627232, 1.13316767003147058429016751628, 1.58337647117500837594788947961, 1.63948706030288368478846581827, 1.90708784257719209813038225991, 1.97769646067554746484348826782, 2.06970862241595621924164299179, 2.12306927751757402615278975838, 2.78063666685823335185512965452, 2.93237805403089410064888819028, 3.01896079649014215695816029054, 3.21576370907211963372992069231, 3.64271520530884667706869018013, 3.79189537710122309498654629439, 3.95344568614688764481435283420, 3.96050267331164881936393987037, 3.97516071198196436787025068960, 4.14102312252380929625495713992, 4.66193978374645556770654987963, 4.66636934234438052566677196502, 4.88516677014278220127961736965, 5.12831259798362014630988071909, 5.18225539372939554376529407600
Plot not available for L-functions of degree greater than 10.
