| L(s) = 1 | + 24·7-s − 12·9-s − 192·23-s + 32·25-s + 144·41-s − 192·47-s + 16·49-s − 288·63-s + 90·81-s − 144·89-s + 72·103-s − 160·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4.60e3·161-s + 163-s + 167-s − 992·169-s + 173-s + 768·175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 24/7·7-s − 4/3·9-s − 8.34·23-s + 1.27·25-s + 3.51·41-s − 4.08·47-s + 0.326·49-s − 4.57·63-s + 10/9·81-s − 1.61·89-s + 0.699·103-s − 1.32·121-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 − 28.6·161-s + 0.00613·163-s + 0.00598·167-s − 5.86·169-s + 0.00578·173-s + 4.38·175-s + 0.00558·179-s + 0.00552·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.375811073\times10^{-6}\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.375811073\times10^{-6}\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p T^{2} )^{4} \) |
| 5 | \( 1 - 32 T^{2} + 6 p^{3} T^{4} - 32 p^{4} T^{6} + p^{8} T^{8} \) |
| good | 7 | \( ( 1 - 6 T + 86 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 + 80 T^{2} + 11982 T^{4} + 80 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 496 T^{2} + 111822 T^{4} + 496 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 328 T^{2} + 23838 T^{4} - 328 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 + 290 T^{2} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 + 24 T + p^{2} T^{2} )^{8} \) |
| 29 | \( ( 1 - 2864 T^{2} + 3458382 T^{4} - 2864 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 2300 T^{2} + 3121158 T^{4} - 2300 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4288 T^{2} + 8283822 T^{4} + 4288 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 36 T + 662 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 - 6676 T^{2} + 17870982 T^{4} - 6676 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 48 T + 1970 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 53 | \( ( 1 + 10384 T^{2} + 42646350 T^{4} + 10384 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 8864 T^{2} + 40876782 T^{4} + 8864 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 + 4124 T^{2} + 16267110 T^{4} + 4124 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 8020 T^{2} + 31887942 T^{4} - 8020 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 7058 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 - 2212 T^{2} + 45633414 T^{4} - 2212 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 2108 T^{2} - 46835706 T^{4} - 2108 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 25876 T^{2} + 261715782 T^{4} - 25876 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 36 T + 4070 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 - 33364 T^{2} + 455045286 T^{4} - 33364 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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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
−4.02367968074451867109428157081, −3.95162827771210956064649499664, −3.82865652880731499502138909521, −3.78891033407257303656407164774, −3.72905091758465551486624786296, −3.56827416586262076375825351313, −3.18648588538343523863501252055, −3.01931943811137150042695005543, −2.93854790363214665690617301907, −2.88553649439195762688543387694, −2.66721798453498855840746549465, −2.41444603893640403015487627129, −2.30983469578774758710312866286, −2.15165743163643032651431336160, −2.08900617646631139884541928224, −1.84177829736740290860433733032, −1.70446541050821082343227499935, −1.67422211712869150076717936323, −1.46902788912707761144656256385, −1.32350056570763926160272140477, −1.23388916234391916951711098978, −0.846719092367898163319440922903, −0.48920124935623255096462357224, −0.05746125831491191508478487654, −0.00055848226386236437406287666,
0.00055848226386236437406287666, 0.05746125831491191508478487654, 0.48920124935623255096462357224, 0.846719092367898163319440922903, 1.23388916234391916951711098978, 1.32350056570763926160272140477, 1.46902788912707761144656256385, 1.67422211712869150076717936323, 1.70446541050821082343227499935, 1.84177829736740290860433733032, 2.08900617646631139884541928224, 2.15165743163643032651431336160, 2.30983469578774758710312866286, 2.41444603893640403015487627129, 2.66721798453498855840746549465, 2.88553649439195762688543387694, 2.93854790363214665690617301907, 3.01931943811137150042695005543, 3.18648588538343523863501252055, 3.56827416586262076375825351313, 3.72905091758465551486624786296, 3.78891033407257303656407164774, 3.82865652880731499502138909521, 3.95162827771210956064649499664, 4.02367968074451867109428157081
Plot not available for L-functions of degree greater than 10.
