| L(s) = 1 | − 60·9-s + 640·23-s − 392·25-s + 984·49-s + 640·71-s − 1.23e3·73-s + 1.24e3·81-s + 5.71e3·97-s + 6.21e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9.06e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 3.84e4·207-s + ⋯ |
| L(s) = 1 | − 2.22·9-s + 5.80·23-s − 3.13·25-s + 2.86·49-s + 1.06·71-s − 1.97·73-s + 1.70·81-s + 5.97·97-s + 4.67·121-s + 0.000698·127-s + 0.000666·131-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 + 4.12·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s − 12.8·207-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(16.54592366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.54592366\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + 10 p T^{2} + p^{6} T^{4} )^{2} \) |
| good | 5 | \( ( 1 + 196 T^{2} + 774 p^{2} T^{4} + 196 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 7 | \( ( 1 - 492 T^{2} + 102278 T^{4} - 492 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 11 | \( ( 1 - 3108 T^{2} + 5613974 T^{4} - 3108 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 13 | \( ( 1 - 4532 T^{2} + 10573590 T^{4} - 4532 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 17 | \( ( 1 - 6916 T^{2} + 54727878 T^{4} - 6916 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 19 | \( ( 1 - 3588 T^{2} + 29873654 T^{4} - 3588 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 23 | \( ( 1 - 160 T + 29390 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 29 | \( ( 1 + 28900 T^{2} + 841043958 T^{4} + 28900 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 - 112780 T^{2} + 4945356198 T^{4} - 112780 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 - 40148 T^{2} + 1484010870 T^{4} - 40148 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 41 | \( ( 1 - 198884 T^{2} + 18596196390 T^{4} - 198884 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 43 | \( ( 1 + 174108 T^{2} + 16916734358 T^{4} + 174108 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 47 | \( ( 1 + 121630 T^{2} + p^{6} T^{4} )^{4} \) |
| 53 | \( ( 1 + 133508 T^{2} - 4558541226 T^{4} + 133508 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 59 | \( ( 1 - 408658 T^{2} + p^{6} T^{4} )^{4} \) |
| 61 | \( ( 1 - 129524 T^{2} + 106016792790 T^{4} - 129524 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 + 690876 T^{2} + 254657072438 T^{4} + 690876 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 71 | \( ( 1 - 160 T - 117778 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 73 | \( ( 1 + 308 T + 608214 T^{2} + 308 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 79 | \( ( 1 - 257100 T^{2} + 501931869158 T^{4} - 257100 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 83 | \( ( 1 - 1153988 T^{2} + 779081408118 T^{4} - 1153988 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 - 1957732 T^{2} + 1892598888294 T^{4} - 1957732 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 97 | \( ( 1 - 1428 T + 2249126 T^{2} - 1428 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
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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
−4.56163521634712613272520153953, −4.27224305671570967695660865122, −4.19486153067814323169476375310, −4.13699698210473260834065319880, −4.04729183365344449767308475544, −3.90179297769442405843463344488, −3.42543556098329336507812365572, −3.28179800491186981315394675913, −3.17641306744327617180398276441, −3.16339787410327835578000707098, −3.02627174009408840717510438605, −2.99849489829863590473139138468, −2.84504096884240121136771618190, −2.36839393053158856146392438205, −2.18819344590808304839282015034, −2.15768913745038328910112456723, −2.09187824509514195469510534669, −1.71950679098176083675573984777, −1.59557313123118151112142879521, −1.18551483360187471304988402036, −1.02340462454041934810866025292, −0.64821897519563936907921809752, −0.52151502239872159293713366760, −0.51619738184525490052247066138, −0.42779299956063003281643348859,
0.42779299956063003281643348859, 0.51619738184525490052247066138, 0.52151502239872159293713366760, 0.64821897519563936907921809752, 1.02340462454041934810866025292, 1.18551483360187471304988402036, 1.59557313123118151112142879521, 1.71950679098176083675573984777, 2.09187824509514195469510534669, 2.15768913745038328910112456723, 2.18819344590808304839282015034, 2.36839393053158856146392438205, 2.84504096884240121136771618190, 2.99849489829863590473139138468, 3.02627174009408840717510438605, 3.16339787410327835578000707098, 3.17641306744327617180398276441, 3.28179800491186981315394675913, 3.42543556098329336507812365572, 3.90179297769442405843463344488, 4.04729183365344449767308475544, 4.13699698210473260834065319880, 4.19486153067814323169476375310, 4.27224305671570967695660865122, 4.56163521634712613272520153953
Plot not available for L-functions of degree greater than 10.
