L(s) = 1 | + 16·7-s − 116·13-s + 448·19-s − 58·25-s − 368·31-s + 1.33e3·37-s − 656·43-s − 44·49-s − 1.00e3·61-s − 320·67-s + 2.44e3·73-s + 64·79-s − 1.85e3·91-s − 2.02e3·97-s + 3.56e3·103-s + 4.26e3·109-s + 3.95e3·121-s + 127-s + 131-s + 7.16e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | + 0.863·7-s − 2.47·13-s + 5.40·19-s − 0.463·25-s − 2.13·31-s + 5.93·37-s − 2.32·43-s − 0.128·49-s − 2.10·61-s − 0.583·67-s + 3.91·73-s + 0.0911·79-s − 2.13·91-s − 2.11·97-s + 3.41·103-s + 3.74·109-s + 2.97·121-s + 0.000698·127-s + 0.000666·131-s + 4.67·133-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(6.893040380\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.893040380\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 58 T^{2} - 1511 T^{4} - 12238 p^{3} T^{6} - 17468 p^{6} T^{8} - 12238 p^{9} T^{10} - 1511 p^{12} T^{12} + 58 p^{18} T^{14} + p^{24} T^{16} \) |
| 7 | \( ( 1 - 8 T + 118 T^{2} + 5920 T^{3} - 136685 T^{4} + 5920 p^{3} T^{5} + 118 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 11 | \( 1 - 3956 T^{2} + 784526 p T^{4} - 13755122768 T^{6} + 18419261375683 T^{8} - 13755122768 p^{6} T^{10} + 784526 p^{13} T^{12} - 3956 p^{18} T^{14} + p^{24} T^{16} \) |
| 13 | \( ( 1 + 58 T - 1115 T^{2} + 4930 T^{3} + 7843924 T^{4} + 4930 p^{3} T^{5} - 1115 p^{6} T^{6} + 58 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 17 | \( ( 1 + 8510 T^{2} + 47982147 T^{4} + 8510 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 19 | \( ( 1 - 112 T + 16098 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 23 | \( 1 - 892 p T^{2} + 104957290 T^{4} - 17730451696 p T^{6} + 15599892891827251 T^{8} - 17730451696 p^{7} T^{10} + 104957290 p^{12} T^{12} - 892 p^{19} T^{14} + p^{24} T^{16} \) |
| 29 | \( 1 - 33062 T^{2} + 284039641 T^{4} + 12583081094218 T^{6} - 349441743692069756 T^{8} + 12583081094218 p^{6} T^{10} + 284039641 p^{12} T^{12} - 33062 p^{18} T^{14} + p^{24} T^{16} \) |
| 31 | \( ( 1 + 184 T - 6974 T^{2} - 3450368 T^{3} - 229018877 T^{4} - 3450368 p^{3} T^{5} - 6974 p^{6} T^{6} + 184 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 - 334 T + 92151 T^{2} - 334 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 41 | \( 1 - 104324 T^{2} + 4124774314 T^{4} + 286002766685680 T^{6} - 28580903421142095053 T^{8} + 286002766685680 p^{6} T^{10} + 4124774314 p^{12} T^{12} - 104324 p^{18} T^{14} + p^{24} T^{16} \) |
| 43 | \( ( 1 + 328 T - 17090 T^{2} - 11263520 T^{3} + 668562139 T^{4} - 11263520 p^{3} T^{5} - 17090 p^{6} T^{6} + 328 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 47 | \( ( 1 - 110446 T^{2} + 1419103587 T^{4} - 110446 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 53 | \( ( 1 + 101876 T^{2} - 13986441354 T^{4} + 101876 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 59 | \( 1 - 467852 T^{2} + 82589971546 T^{4} - 24297638676216752 T^{6} + \)\(75\!\cdots\!19\)\( T^{8} - 24297638676216752 p^{6} T^{10} + 82589971546 p^{12} T^{12} - 467852 p^{18} T^{14} + p^{24} T^{16} \) |
| 61 | \( ( 1 + 502 T - 203723 T^{2} + 886030 T^{3} + 98966604244 T^{4} + 886030 p^{3} T^{5} - 203723 p^{6} T^{6} + 502 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 + 160 T - 412226 T^{2} - 26192000 T^{3} + 93533691307 T^{4} - 26192000 p^{3} T^{5} - 412226 p^{6} T^{6} + 160 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 71 | \( ( 1 + 625028 T^{2} + 209892928038 T^{4} + 625028 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 - 305 T + p^{3} T^{2} )^{8} \) |
| 79 | \( ( 1 - 32 T - 512810 T^{2} + 15111808 T^{3} + 20433052099 T^{4} + 15111808 p^{3} T^{5} - 512810 p^{6} T^{6} - 32 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 83 | \( 1 - 436460 T^{2} + 48969507898 T^{4} + 223621556788292560 T^{6} - \)\(14\!\cdots\!57\)\( T^{8} + 223621556788292560 p^{6} T^{10} + 48969507898 p^{12} T^{12} - 436460 p^{18} T^{14} + p^{24} T^{16} \) |
| 89 | \( ( 1 - 1173874 T^{2} + 984168880467 T^{4} - 1173874 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 97 | \( ( 1 + 1012 T - 1045142 T^{2} + 246867280 T^{3} + 2514263967955 T^{4} + 246867280 p^{3} T^{5} - 1045142 p^{6} T^{6} + 1012 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.61558866340450378740636190479, −4.61404629119002508867523852235, −4.60701203418665460800181005790, −4.46953687994398689298633005975, −3.94193175218541128251614568552, −3.77789027186043707073241908151, −3.73840600205194761409173401132, −3.64669124855516786714526498745, −3.52986911218143326055698084780, −3.14434748082294262273910130223, −3.02091594033821202734943492866, −3.00422289412848975349956309640, −2.70763001227739790006868821025, −2.67407058673458532971182270584, −2.29611079768222233521721746126, −2.21095296083330576836607996820, −2.15711990184551816233934113612, −1.75643752160617775488624491633, −1.42362076257383369611857250900, −1.41366243496511077079417644923, −1.05670245107132424457929804283, −0.901487896417828839303380766430, −0.856095681072410385770658158573, −0.32236806179790111012659929934, −0.26070332569927542087769649963,
0.26070332569927542087769649963, 0.32236806179790111012659929934, 0.856095681072410385770658158573, 0.901487896417828839303380766430, 1.05670245107132424457929804283, 1.41366243496511077079417644923, 1.42362076257383369611857250900, 1.75643752160617775488624491633, 2.15711990184551816233934113612, 2.21095296083330576836607996820, 2.29611079768222233521721746126, 2.67407058673458532971182270584, 2.70763001227739790006868821025, 3.00422289412848975349956309640, 3.02091594033821202734943492866, 3.14434748082294262273910130223, 3.52986911218143326055698084780, 3.64669124855516786714526498745, 3.73840600205194761409173401132, 3.77789027186043707073241908151, 3.94193175218541128251614568552, 4.46953687994398689298633005975, 4.60701203418665460800181005790, 4.61404629119002508867523852235, 4.61558866340450378740636190479
Plot not available for L-functions of degree greater than 10.