| L(s) = 1 | − 81·9-s − 1.12e3·11-s + 5.72e3·19-s − 2.26e3·29-s − 1.20e4·31-s + 2.27e4·41-s + 3.33e4·49-s + 5.23e4·59-s + 1.88e4·61-s + 2.90e4·71-s + 1.94e5·79-s + 6.56e3·81-s + 9.58e4·89-s + 9.13e4·99-s + 1.70e5·101-s + 2.93e5·109-s + 6.32e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6.05e5·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 2.81·11-s + 3.63·19-s − 0.500·29-s − 2.24·31-s + 2.11·41-s + 1.98·49-s + 1.95·59-s + 0.648·61-s + 0.683·71-s + 3.50·79-s + 1/9·81-s + 1.28·89-s + 0.936·99-s + 1.66·101-s + 2.36·109-s + 3.92·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.63·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.407884221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.407884221\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 33358 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 564 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 605686 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 1660318 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2860 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10363630 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 1134 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6016 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 138398470 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 11370 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 264379750 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 352682398 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 371727578 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 444 p T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 9422 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 86586838 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 14520 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3631851502 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 97312 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7814783350 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 47910 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2632504130 T^{2} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.97110092408983469465050690170, −10.82496157060521223329901050987, −10.12389773340688517323163546642, −9.849607469096474045812989396319, −9.141965045630881007810171950649, −9.010331075890822638387287322738, −7.907091319559912372276657153289, −7.83178075360141745277805124108, −7.38993262570852929343217259988, −7.08055740612991876564117542543, −5.86071725854605011670080534104, −5.57422256013872208556669576378, −5.22622089462969165116924676058, −4.86992263637292555453780463350, −3.51841792855409096749709270200, −3.50902051849129767067991588583, −2.37642269413640562830914766148, −2.37418720420852210045740984550, −0.942628887434728886593461104622, −0.51624678140869409620760055258,
0.51624678140869409620760055258, 0.942628887434728886593461104622, 2.37418720420852210045740984550, 2.37642269413640562830914766148, 3.50902051849129767067991588583, 3.51841792855409096749709270200, 4.86992263637292555453780463350, 5.22622089462969165116924676058, 5.57422256013872208556669576378, 5.86071725854605011670080534104, 7.08055740612991876564117542543, 7.38993262570852929343217259988, 7.83178075360141745277805124108, 7.907091319559912372276657153289, 9.010331075890822638387287322738, 9.141965045630881007810171950649, 9.849607469096474045812989396319, 10.12389773340688517323163546642, 10.82496157060521223329901050987, 10.97110092408983469465050690170
