| L(s) = 1 | + 136·7-s − 2.18e3·9-s − 2.58e4·17-s + 1.26e4·23-s + 2.25e5·25-s + 2.20e5·31-s − 6.19e5·41-s − 3.26e5·47-s − 2.75e6·49-s − 2.97e5·63-s + 5.84e6·71-s − 1.10e7·73-s + 4.95e5·79-s + 3.18e6·81-s + 9.66e6·89-s − 9.51e6·97-s − 3.31e7·103-s + 1.80e7·113-s − 3.50e6·119-s + 5.45e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 5.64e7·153-s + ⋯ |
| L(s) = 1 | + 0.149·7-s − 9-s − 1.27·17-s + 0.217·23-s + 2.88·25-s + 1.32·31-s − 1.40·41-s − 0.458·47-s − 3.34·49-s − 0.149·63-s + 1.93·71-s − 3.33·73-s + 0.113·79-s + 2/3·81-s + 1.45·89-s − 1.05·97-s − 2.99·103-s + 1.17·113-s − 0.190·119-s + 2.79·121-s + 1.27·153-s + 0.0325·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 3^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 3^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.033364129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.033364129\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{6} T^{2} )^{3} \) |
| good | 5 | \( 1 - 225166 T^{2} + 1252259007 p^{2} T^{4} - 4750796153444 p^{4} T^{6} + 1252259007 p^{16} T^{8} - 225166 p^{28} T^{10} + p^{42} T^{12} \) |
| 7 | \( ( 1 - 68 T + 197571 p T^{2} + 241496840 T^{3} + 197571 p^{8} T^{4} - 68 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 11 | \( 1 - 54520034 T^{2} + 1201965465160903 T^{4} - \)\(20\!\cdots\!04\)\( T^{6} + 1201965465160903 p^{14} T^{8} - 54520034 p^{28} T^{10} + p^{42} T^{12} \) |
| 13 | \( 1 - 238033326 T^{2} + 30084594220983543 T^{4} - \)\(23\!\cdots\!60\)\( T^{6} + 30084594220983543 p^{14} T^{8} - 238033326 p^{28} T^{10} + p^{42} T^{12} \) |
| 17 | \( ( 1 + 12902 T + 579576287 T^{2} + 7165169783700 T^{3} + 579576287 p^{7} T^{4} + 12902 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 19 | \( 1 - 2542066226 T^{2} + 3322401748815979319 T^{4} - \)\(86\!\cdots\!24\)\( p^{2} T^{6} + 3322401748815979319 p^{14} T^{8} - 2542066226 p^{28} T^{10} + p^{42} T^{12} \) |
| 23 | \( ( 1 - 6344 T + 9273900021 T^{2} - 48873638231792 T^{3} + 9273900021 p^{7} T^{4} - 6344 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 29 | \( 1 - 16184310462 T^{2} + \)\(33\!\cdots\!59\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{6} + \)\(33\!\cdots\!59\)\( p^{14} T^{8} - 16184310462 p^{28} T^{10} + p^{42} T^{12} \) |
| 31 | \( ( 1 - 3556 p T + 55124926221 T^{2} - 3430233756506120 T^{3} + 55124926221 p^{7} T^{4} - 3556 p^{15} T^{5} + p^{21} T^{6} )^{2} \) |
| 37 | \( 1 - 427100858270 T^{2} + \)\(82\!\cdots\!31\)\( T^{4} - \)\(97\!\cdots\!16\)\( T^{6} + \)\(82\!\cdots\!31\)\( p^{14} T^{8} - 427100858270 p^{28} T^{10} + p^{42} T^{12} \) |
| 41 | \( ( 1 + 309826 T + 319548055735 T^{2} + 84007116295570300 T^{3} + 319548055735 p^{7} T^{4} + 309826 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 43 | \( 1 - 918223611938 T^{2} + \)\(45\!\cdots\!35\)\( T^{4} - \)\(15\!\cdots\!08\)\( T^{6} + \)\(45\!\cdots\!35\)\( p^{14} T^{8} - 918223611938 p^{28} T^{10} + p^{42} T^{12} \) |
| 47 | \( ( 1 + 163016 T + 849521362605 T^{2} + 290579616817963376 T^{3} + 849521362605 p^{7} T^{4} + 163016 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 53 | \( 1 - 5108541325998 T^{2} + \)\(12\!\cdots\!07\)\( T^{4} - \)\(17\!\cdots\!24\)\( T^{6} + \)\(12\!\cdots\!07\)\( p^{14} T^{8} - 5108541325998 p^{28} T^{10} + p^{42} T^{12} \) |
| 59 | \( 1 - 7539226325058 T^{2} + \)\(34\!\cdots\!03\)\( T^{4} - \)\(29\!\cdots\!32\)\( p^{2} T^{6} + \)\(34\!\cdots\!03\)\( p^{14} T^{8} - 7539226325058 p^{28} T^{10} + p^{42} T^{12} \) |
| 61 | \( 1 - 8221966027278 T^{2} + \)\(35\!\cdots\!51\)\( T^{4} - \)\(12\!\cdots\!72\)\( T^{6} + \)\(35\!\cdots\!51\)\( p^{14} T^{8} - 8221966027278 p^{28} T^{10} + p^{42} T^{12} \) |
| 67 | \( 1 - 5173503552402 T^{2} + \)\(11\!\cdots\!95\)\( T^{4} - \)\(37\!\cdots\!72\)\( T^{6} + \)\(11\!\cdots\!95\)\( p^{14} T^{8} - 5173503552402 p^{28} T^{10} + p^{42} T^{12} \) |
| 71 | \( ( 1 - 2920888 T + 230965942899 p T^{2} - 770831665799538800 p T^{3} + 230965942899 p^{8} T^{4} - 2920888 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 73 | \( ( 1 + 5537598 T + 29690123235159 T^{2} + \)\(11\!\cdots\!56\)\( T^{3} + 29690123235159 p^{7} T^{4} + 5537598 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 79 | \( ( 1 - 247900 T + 43961623042557 T^{2} + 11061094914268952056 T^{3} + 43961623042557 p^{7} T^{4} - 247900 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 83 | \( 1 - 46865850155186 T^{2} + \)\(16\!\cdots\!79\)\( T^{4} - \)\(68\!\cdots\!92\)\( p^{2} T^{6} + \)\(16\!\cdots\!79\)\( p^{14} T^{8} - 46865850155186 p^{28} T^{10} + p^{42} T^{12} \) |
| 89 | \( ( 1 - 4830370 T + 21352026366247 T^{2} + \)\(23\!\cdots\!92\)\( T^{3} + 21352026366247 p^{7} T^{4} - 4830370 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 97 | \( ( 1 + 4755782 T + 231060382517487 T^{2} + \)\(77\!\cdots\!28\)\( T^{3} + 231060382517487 p^{7} T^{4} + 4755782 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
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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
−4.92901639711799479414250386265, −4.78225477718286856899687863886, −4.67720915196792102503390390253, −4.60531734624067992015622144444, −4.13707548078428851141252998448, −4.10148804270459475689066053916, −3.58424561752324096155816378156, −3.55072008370034317208959758152, −3.35655948955778841451148416994, −3.34203681963633321252953806823, −3.12268088835395808816604323049, −2.74208643549759955293272895588, −2.53500342971031941010919880021, −2.49615040952665279326536645753, −2.38640057063543077218116224258, −2.15750440868663689459330443578, −1.82133618961408848401421265961, −1.43908167481386305075220104253, −1.30498134153901688934995939362, −1.28904448936093639153097395334, −1.01109661203314993433212472091, −0.940365306979325588257905652021, −0.32108612971596459784882487334, −0.27288863584751004556411276886, −0.13249593894147130001226001420,
0.13249593894147130001226001420, 0.27288863584751004556411276886, 0.32108612971596459784882487334, 0.940365306979325588257905652021, 1.01109661203314993433212472091, 1.28904448936093639153097395334, 1.30498134153901688934995939362, 1.43908167481386305075220104253, 1.82133618961408848401421265961, 2.15750440868663689459330443578, 2.38640057063543077218116224258, 2.49615040952665279326536645753, 2.53500342971031941010919880021, 2.74208643549759955293272895588, 3.12268088835395808816604323049, 3.34203681963633321252953806823, 3.35655948955778841451148416994, 3.55072008370034317208959758152, 3.58424561752324096155816378156, 4.10148804270459475689066053916, 4.13707548078428851141252998448, 4.60531734624067992015622144444, 4.67720915196792102503390390253, 4.78225477718286856899687863886, 4.92901639711799479414250386265
Plot not available for L-functions of degree greater than 10.
