| L(s) = 1 | + 15·9-s − 72·11-s − 4·19-s + 126·29-s − 14·31-s − 36·41-s − 47·49-s − 252·59-s + 82·61-s − 166·79-s + 81·81-s − 1.08e3·99-s + 882·101-s − 664·109-s + 2.52e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 515·169-s − 60·171-s + 173-s + ⋯ |
| L(s) = 1 | + 5/3·9-s − 6.54·11-s − 0.210·19-s + 4.34·29-s − 0.451·31-s − 0.878·41-s − 0.959·49-s − 4.27·59-s + 1.34·61-s − 2.10·79-s + 81-s − 10.9·99-s + 8.73·101-s − 6.09·109-s + 20.8·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 3.04·169-s − 0.350·171-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3964622171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3964622171\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 5 p T^{2} + 16 p^{2} T^{4} - 5 p^{5} T^{6} + p^{8} T^{8} \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 47 T^{2} - 3071 T^{4} + 22466 T^{6} + 16809790 T^{8} + 22466 p^{4} T^{10} - 3071 p^{8} T^{12} + 47 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( ( 1 + 36 T + 683 T^{2} + 9036 T^{3} + 100632 T^{4} + 9036 p^{2} T^{5} + 683 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( 1 + 515 T^{2} + 143653 T^{4} + 33191750 T^{6} + 6388929238 T^{8} + 33191750 p^{4} T^{10} + 143653 p^{8} T^{12} + 515 p^{12} T^{14} + p^{16} T^{16} \) |
| 17 | \( ( 1 + 769 T^{2} + 298176 T^{4} + 769 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 + T + 648 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( 1 - 433 T^{2} - 338207 T^{4} + 14715938 T^{6} + 145578189886 T^{8} + 14715938 p^{4} T^{10} - 338207 p^{8} T^{12} - 433 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 - 63 T + 2123 T^{2} - 50400 T^{3} + 1045362 T^{4} - 50400 p^{2} T^{5} + 2123 p^{4} T^{6} - 63 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 + 7 T - 1217 T^{2} - 4592 T^{3} + 632146 T^{4} - 4592 p^{2} T^{5} - 1217 p^{4} T^{6} + 7 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2588 T^{2} + 4206246 T^{4} - 2588 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 18 T + 1913 T^{2} + 32490 T^{3} + 613812 T^{4} + 32490 p^{2} T^{5} + 1913 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 3169 T^{2} + 6623760 T^{4} + 3169 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 - 7297 T^{2} + 30662449 T^{4} - 93579632206 T^{6} + 226390414696510 T^{8} - 93579632206 p^{4} T^{10} + 30662449 p^{8} T^{12} - 7297 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 + 7204 T^{2} + 26018214 T^{4} + 7204 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 126 T + 11993 T^{2} + 844326 T^{3} + 51207492 T^{4} + 844326 p^{2} T^{5} + 11993 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 41 T - 5513 T^{2} + 10168 T^{3} + 31652794 T^{4} + 10168 p^{2} T^{5} - 5513 p^{4} T^{6} - 41 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 + 5882 T^{2} + 21614089 T^{4} - 160686869974 T^{6} - 934674659122220 T^{8} - 160686869974 p^{4} T^{10} + 21614089 p^{8} T^{12} + 5882 p^{12} T^{14} + p^{16} T^{16} \) |
| 71 | \( ( 1 - 18616 T^{2} + 137194926 T^{4} - 18616 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 19055 T^{2} + 146334144 T^{4} - 19055 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 83 T - 5459 T^{2} - 11122 T^{3} + 70528774 T^{4} - 11122 p^{2} T^{5} - 5459 p^{4} T^{6} + 83 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 - 26017 T^{2} + 413233729 T^{4} - 4389950344606 T^{6} + 35135533688955790 T^{8} - 4389950344606 p^{4} T^{10} + 413233729 p^{8} T^{12} - 26017 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 - 6916 T^{2} + 69013446 T^{4} - 6916 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( 1 + 17834 T^{2} + 72889657 T^{4} + 1214554912058 T^{6} + 24110047553298004 T^{8} + 1214554912058 p^{4} T^{10} + 72889657 p^{8} T^{12} + 17834 p^{12} T^{14} + p^{16} T^{16} \) |
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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
−4.21966635286610208204017709909, −4.09096369349225989902332625733, −4.02231942661122222288609636105, −3.58593099893320078981380348802, −3.42598857852000877229716246306, −3.38325245664804263418409264853, −3.09101016457065934598515537311, −3.07749028896389917170516973700, −3.03104164405342558640170424172, −2.98191191907606692152093732800, −2.77191176902965249641059941206, −2.58573248861260044724074813166, −2.36743091357589865182625146099, −2.33601623303902940849505524529, −2.13085709399936341597146129830, −2.07881307322264645270763058692, −1.88068127901219124197297961190, −1.65763426479357126603938224908, −1.35035556493819475310231751458, −1.11523283920814931649902197249, −1.09382206351153583528933093801, −0.822324429850365940067472494495, −0.36930291981909113882719171689, −0.34802438464279193018051504883, −0.087485632571922983782940885582,
0.087485632571922983782940885582, 0.34802438464279193018051504883, 0.36930291981909113882719171689, 0.822324429850365940067472494495, 1.09382206351153583528933093801, 1.11523283920814931649902197249, 1.35035556493819475310231751458, 1.65763426479357126603938224908, 1.88068127901219124197297961190, 2.07881307322264645270763058692, 2.13085709399936341597146129830, 2.33601623303902940849505524529, 2.36743091357589865182625146099, 2.58573248861260044724074813166, 2.77191176902965249641059941206, 2.98191191907606692152093732800, 3.03104164405342558640170424172, 3.07749028896389917170516973700, 3.09101016457065934598515537311, 3.38325245664804263418409264853, 3.42598857852000877229716246306, 3.58593099893320078981380348802, 4.02231942661122222288609636105, 4.09096369349225989902332625733, 4.21966635286610208204017709909
Plot not available for L-functions of degree greater than 10.
