| L(s) = 1 | − 2·5-s + 108·11-s − 228·19-s + 113·25-s − 260·29-s − 60·31-s + 12·41-s − 147·49-s − 216·55-s − 1.23e3·59-s − 276·61-s + 948·71-s + 1.22e3·79-s − 4.05e3·89-s + 456·95-s + 5.29e3·101-s + 1.42e3·109-s + 450·121-s + 216·125-s + 127-s + 131-s + 137-s + 139-s + 520·145-s + 149-s + 151-s + 120·155-s + ⋯ |
| L(s) = 1 | − 0.178·5-s + 2.96·11-s − 2.75·19-s + 0.903·25-s − 1.66·29-s − 0.347·31-s + 0.0457·41-s − 3/7·49-s − 0.529·55-s − 2.71·59-s − 0.579·61-s + 1.58·71-s + 1.74·79-s − 4.82·89-s + 0.492·95-s + 5.21·101-s + 1.25·109-s + 0.338·121-s + 0.154·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.297·145-s + 0.000549·149-s + 0.000538·151-s + 0.0621·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.813466602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.813466602\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2 T - 109 T^{2} - 132 p T^{3} - 109 p^{3} T^{4} + 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | \( ( 1 + p^{2} T^{2} )^{3} \) |
| good | 11 | \( ( 1 - 54 T + 4149 T^{2} - 143532 T^{3} + 4149 p^{3} T^{4} - 54 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 13 | \( 1 - 8850 T^{2} + 35538327 T^{4} - 91902094300 T^{6} + 35538327 p^{6} T^{8} - 8850 p^{12} T^{10} + p^{18} T^{12} \) |
| 17 | \( 1 - 25350 T^{2} + 285909807 T^{4} - 1820971953300 T^{6} + 285909807 p^{6} T^{8} - 25350 p^{12} T^{10} + p^{18} T^{12} \) |
| 19 | \( ( 1 + 6 p T + 20733 T^{2} + 1491076 T^{3} + 20733 p^{3} T^{4} + 6 p^{7} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 - 33738 T^{2} + 719506527 T^{4} - 10503331546764 T^{6} + 719506527 p^{6} T^{8} - 33738 p^{12} T^{10} + p^{18} T^{12} \) |
| 29 | \( ( 1 + 130 T + 38219 T^{2} + 7786188 T^{3} + 38219 p^{3} T^{4} + 130 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( ( 1 + 30 T + 87801 T^{2} + 1741004 T^{3} + 87801 p^{3} T^{4} + 30 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 37 | \( 1 - 213390 T^{2} + 20725249095 T^{4} - 1264461023575204 T^{6} + 20725249095 p^{6} T^{8} - 213390 p^{12} T^{10} + p^{18} T^{12} \) |
| 41 | \( ( 1 - 6 T + 154791 T^{2} - 4441812 T^{3} + 154791 p^{3} T^{4} - 6 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( 1 - 406098 T^{2} + 73852437927 T^{4} - 7603035363807004 T^{6} + 73852437927 p^{6} T^{8} - 406098 p^{12} T^{10} + p^{18} T^{12} \) |
| 47 | \( 1 - 342858 T^{2} + 58157280879 T^{4} - 6800396023555404 T^{6} + 58157280879 p^{6} T^{8} - 342858 p^{12} T^{10} + p^{18} T^{12} \) |
| 53 | \( 1 - 44802 T^{2} + 9565330599 T^{4} - 6431440620065596 T^{6} + 9565330599 p^{6} T^{8} - 44802 p^{12} T^{10} + p^{18} T^{12} \) |
| 59 | \( ( 1 + 616 T + 288905 T^{2} + 108456048 T^{3} + 288905 p^{3} T^{4} + 616 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 61 | \( ( 1 + 138 T + 115755 T^{2} - 55635204 T^{3} + 115755 p^{3} T^{4} + 138 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 67 | \( 1 - 206802 T^{2} + 210864306807 T^{4} - 40454618849562076 T^{6} + 210864306807 p^{6} T^{8} - 206802 p^{12} T^{10} + p^{18} T^{12} \) |
| 71 | \( ( 1 - 474 T + 752913 T^{2} - 234005028 T^{3} + 752913 p^{3} T^{4} - 474 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( 1 - 1243914 T^{2} + 879311853087 T^{4} - 400903520866190092 T^{6} + 879311853087 p^{6} T^{8} - 1243914 p^{12} T^{10} + p^{18} T^{12} \) |
| 79 | \( ( 1 - 612 T + 1596477 T^{2} - 610671736 T^{3} + 1596477 p^{3} T^{4} - 612 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 83 | \( 1 - 1184706 T^{2} + 801806844951 T^{4} - 498093185721850108 T^{6} + 801806844951 p^{6} T^{8} - 1184706 p^{12} T^{10} + p^{18} T^{12} \) |
| 89 | \( ( 1 + 2026 T + 3442103 T^{2} + 3135596364 T^{3} + 3442103 p^{3} T^{4} + 2026 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 - 1081146 T^{2} + 1215707220687 T^{4} - 758120958269212268 T^{6} + 1215707220687 p^{6} T^{8} - 1081146 p^{12} T^{10} + p^{18} T^{12} \) |
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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
−4.56290245708112225946906757654, −4.50436876460464942436408501422, −4.48344930866880909153857766298, −4.23763047716792268728813151558, −3.98593377282722676513103045549, −3.81313263379893485309237271382, −3.77383057989660856907509660878, −3.55706057493080754012577692532, −3.49911524409665313492087623097, −3.39542899664315455749771749785, −3.10157044217355739096771329977, −2.67553379982664191088852870976, −2.65067306314120367278332342769, −2.55172518392797590986719419307, −2.21725313560329740800568748168, −2.11936005323865866496640091178, −1.82709204554803583980386673015, −1.62205911116099382757175658573, −1.47526408664153861504683500243, −1.30746157720415127466026636784, −1.22423308595433540869357672444, −0.923275819957428873390585692771, −0.37782774297118591892690888413, −0.37697191031628364862123921653, −0.26924650316125941118006618351,
0.26924650316125941118006618351, 0.37697191031628364862123921653, 0.37782774297118591892690888413, 0.923275819957428873390585692771, 1.22423308595433540869357672444, 1.30746157720415127466026636784, 1.47526408664153861504683500243, 1.62205911116099382757175658573, 1.82709204554803583980386673015, 2.11936005323865866496640091178, 2.21725313560329740800568748168, 2.55172518392797590986719419307, 2.65067306314120367278332342769, 2.67553379982664191088852870976, 3.10157044217355739096771329977, 3.39542899664315455749771749785, 3.49911524409665313492087623097, 3.55706057493080754012577692532, 3.77383057989660856907509660878, 3.81313263379893485309237271382, 3.98593377282722676513103045549, 4.23763047716792268728813151558, 4.48344930866880909153857766298, 4.50436876460464942436408501422, 4.56290245708112225946906757654
Plot not available for L-functions of degree greater than 10.
