| L(s) = 1 | + 126·4-s + 8.77e3·16-s − 4.32e3·19-s + 6.01e4·31-s − 1.06e6·49-s − 4.49e5·61-s + 1.66e5·64-s − 5.44e5·76-s − 4.32e6·79-s − 9.75e6·109-s + 9.52e6·121-s + 7.58e6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.56e7·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 1.96·4-s + 2.14·16-s − 0.629·19-s + 2.02·31-s − 9.02·49-s − 1.98·61-s + 0.633·64-s − 1.23·76-s − 8.77·79-s − 7.52·109-s + 5.37·121-s + 3.97·124-s − 3.23·169-s − 17.7·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+3)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(7.839225224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.839225224\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( ( 1 - 63 T^{2} + 783 p T^{4} + 473 p^{8} T^{6} + 783 p^{13} T^{8} - 63 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 7 | \( ( 1 + 530826 T^{2} + 19160571525 p T^{4} + 2859454341824300 p T^{6} + 19160571525 p^{13} T^{8} + 530826 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 11 | \( ( 1 - 4764642 T^{2} + 137917962411 p^{2} T^{4} - 2284490377309508 p^{4} T^{6} + 137917962411 p^{14} T^{8} - 4764642 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 13 | \( ( 1 + 7804086 T^{2} + 45074199232275 T^{4} + \)\(26\!\cdots\!60\)\( T^{6} + 45074199232275 p^{12} T^{8} + 7804086 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 17 | \( ( 1 - 83837988 T^{2} + 3621200530818951 T^{4} - \)\(10\!\cdots\!92\)\( T^{6} + 3621200530818951 p^{12} T^{8} - 83837988 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 19 | \( ( 1 + 1080 T + 63012123 T^{2} - 28203249040 T^{3} + 63012123 p^{6} T^{4} + 1080 p^{12} T^{5} + p^{18} T^{6} )^{4} \) |
| 23 | \( ( 1 - 331689870 T^{2} + 63881725455493983 T^{4} - \)\(37\!\cdots\!80\)\( p T^{6} + 63881725455493983 p^{12} T^{8} - 331689870 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 29 | \( ( 1 - 2825907540 T^{2} + 3679177215775060023 T^{4} - \)\(27\!\cdots\!80\)\( T^{6} + 3679177215775060023 p^{12} T^{8} - 2825907540 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 31 | \( ( 1 - 15048 T + 2310776631 T^{2} - 25499167028192 T^{3} + 2310776631 p^{6} T^{4} - 15048 p^{12} T^{5} + p^{18} T^{6} )^{4} \) |
| 37 | \( ( 1 - 256870758 T^{2} + 18115473533763880851 T^{4} - \)\(24\!\cdots\!72\)\( T^{6} + 18115473533763880851 p^{12} T^{8} - 256870758 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 41 | \( ( 1 - 19394280960 T^{2} + \)\(18\!\cdots\!63\)\( T^{4} - \)\(11\!\cdots\!20\)\( T^{6} + \)\(18\!\cdots\!63\)\( p^{12} T^{8} - 19394280960 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 43 | \( ( 1 + 17318641542 T^{2} + \)\(15\!\cdots\!91\)\( T^{4} + \)\(10\!\cdots\!28\)\( T^{6} + \)\(15\!\cdots\!91\)\( p^{12} T^{8} + 17318641542 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 47 | \( ( 1 - 25637949030 T^{2} + \)\(25\!\cdots\!43\)\( T^{4} - \)\(20\!\cdots\!60\)\( T^{6} + \)\(25\!\cdots\!43\)\( p^{12} T^{8} - 25637949030 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 53 | \( ( 1 - 85310685540 T^{2} + \)\(36\!\cdots\!03\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{6} + \)\(36\!\cdots\!03\)\( p^{12} T^{8} - 85310685540 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 59 | \( ( 1 - 105474519042 T^{2} + \)\(65\!\cdots\!11\)\( T^{4} - \)\(31\!\cdots\!68\)\( T^{6} + \)\(65\!\cdots\!11\)\( p^{12} T^{8} - 105474519042 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 61 | \( ( 1 + 112446 T + 80495409735 T^{2} + 16896003232280740 T^{3} + 80495409735 p^{6} T^{4} + 112446 p^{12} T^{5} + p^{18} T^{6} )^{4} \) |
| 67 | \( ( 1 + 100915149462 T^{2} + \)\(11\!\cdots\!11\)\( T^{4} + \)\(18\!\cdots\!48\)\( T^{6} + \)\(11\!\cdots\!11\)\( p^{12} T^{8} + 100915149462 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 71 | \( ( 1 - 219093237510 T^{2} + \)\(21\!\cdots\!03\)\( T^{4} - \)\(24\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!03\)\( p^{12} T^{8} - 219093237510 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 73 | \( ( 1 + 408556507506 T^{2} + \)\(11\!\cdots\!95\)\( T^{4} + \)\(19\!\cdots\!00\)\( T^{6} + \)\(11\!\cdots\!95\)\( p^{12} T^{8} + 408556507506 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 79 | \( ( 1 + 1081152 T + 840245103351 T^{2} + 513306877165874768 T^{3} + 840245103351 p^{6} T^{4} + 1081152 p^{12} T^{5} + p^{18} T^{6} )^{4} \) |
| 83 | \( ( 1 - 1277278858710 T^{2} + \)\(81\!\cdots\!03\)\( T^{4} - \)\(33\!\cdots\!20\)\( T^{6} + \)\(81\!\cdots\!03\)\( p^{12} T^{8} - 1277278858710 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 89 | \( ( 1 + 410253320928 T^{2} + \)\(72\!\cdots\!71\)\( T^{4} + \)\(20\!\cdots\!32\)\( T^{6} + \)\(72\!\cdots\!71\)\( p^{12} T^{8} + 410253320928 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 97 | \( ( 1 + 2586932231442 T^{2} + \)\(25\!\cdots\!91\)\( T^{4} + \)\(18\!\cdots\!08\)\( T^{6} + \)\(25\!\cdots\!91\)\( p^{12} T^{8} + 2586932231442 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.97579777005561186906258700266, −2.92494129078431541426831358612, −2.86997432597227075570519139756, −2.77322139246694292805243403519, −2.71582560341836308762524714939, −2.68001911301443795920793473903, −2.45390044024830126009953143207, −2.06276970992676753150382392274, −1.90781034177942094166868696564, −1.87954219787244132166463767407, −1.82484654905329076116896432313, −1.77095614625439953300364052841, −1.72324799712644386956538536646, −1.43521586430125261483872215195, −1.41393145827124688198738196060, −1.39657539876101887516271036317, −1.36051776584467276433973944057, −1.23589043235543277241832072592, −0.71820322315865518910903978849, −0.57198094869902228031253902611, −0.53060147475366928705606970075, −0.52503929901579764741764620813, −0.46375016494423628273285332457, −0.23373797828215977924314956854, −0.080499011027833971635767188208,
0.080499011027833971635767188208, 0.23373797828215977924314956854, 0.46375016494423628273285332457, 0.52503929901579764741764620813, 0.53060147475366928705606970075, 0.57198094869902228031253902611, 0.71820322315865518910903978849, 1.23589043235543277241832072592, 1.36051776584467276433973944057, 1.39657539876101887516271036317, 1.41393145827124688198738196060, 1.43521586430125261483872215195, 1.72324799712644386956538536646, 1.77095614625439953300364052841, 1.82484654905329076116896432313, 1.87954219787244132166463767407, 1.90781034177942094166868696564, 2.06276970992676753150382392274, 2.45390044024830126009953143207, 2.68001911301443795920793473903, 2.71582560341836308762524714939, 2.77322139246694292805243403519, 2.86997432597227075570519139756, 2.92494129078431541426831358612, 2.97579777005561186906258700266
Plot not available for L-functions of degree greater than 10.
