| L(s) = 1 | − 96·19-s + 56·25-s − 160·43-s + 152·49-s − 192·67-s + 304·73-s + 112·97-s − 520·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 312·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 5.05·19-s + 2.23·25-s − 3.72·43-s + 3.10·49-s − 2.86·67-s + 4.16·73-s + 1.15·97-s − 4.29·121-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 + 1.84·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.843211005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.843211005\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 28 T^{2} + 22 p^{2} T^{4} - 28 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 - 76 T^{2} + 5350 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 + 130 T^{2} + p^{4} T^{4} )^{4} \) |
| 13 | \( ( 1 - 12 p T^{2} + 30950 T^{4} - 12 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 868 T^{2} + 341062 T^{4} + 868 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 + 24 T + 642 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 - 196 T^{2} - 348218 T^{4} - 196 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 2524 T^{2} + 2963302 T^{4} - 2524 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 3084 T^{2} + 4152230 T^{4} - 3084 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 5020 T^{2} + 10044838 T^{4} - 5020 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 3364 T^{2} + 7778182 T^{4} + 3364 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 40 T + 3874 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 47 | \( ( 1 + 252 T^{2} + 1517702 T^{4} + 252 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4060 T^{2} + 11815462 T^{4} - 4060 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 8932 T^{2} + 40509862 T^{4} + 8932 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 14044 T^{2} + 76824550 T^{4} - 14044 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 48 T + 1490 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 - 1890 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 - 76 T + 4038 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 17228 T^{2} + 145319334 T^{4} - 17228 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 + 10948 T^{2} + 121211302 T^{4} + 10948 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 11076 T^{2} + 54999110 T^{4} + 11076 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 28 T + 15430 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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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
−3.54020780286309454517647484114, −3.44413110419457538944626556062, −3.38044982396807429428174909737, −3.29990016093481708768365530040, −3.26251046505719505598509328709, −2.87767050212256409319558424249, −2.61775868549872672952302147178, −2.50938415484935028289251819199, −2.45774456996433370157818986476, −2.38924989557373394682596906014, −2.38572041733964948988143809178, −2.26845101979555256413299697067, −2.25598138494463696018257223005, −2.03515288910539254330528981981, −1.68700479650629806455347407208, −1.44984827638128724690912778235, −1.35309770221533826214127098669, −1.34931559078848148991275550338, −1.25378469154265264013508793878, −1.19077855987142597990982206702, −0.837113984727372158133216703375, −0.40124511579113262428514348288, −0.39569347681583617193023228897, −0.23828766475439324093409652771, −0.21229107167831392308300685260,
0.21229107167831392308300685260, 0.23828766475439324093409652771, 0.39569347681583617193023228897, 0.40124511579113262428514348288, 0.837113984727372158133216703375, 1.19077855987142597990982206702, 1.25378469154265264013508793878, 1.34931559078848148991275550338, 1.35309770221533826214127098669, 1.44984827638128724690912778235, 1.68700479650629806455347407208, 2.03515288910539254330528981981, 2.25598138494463696018257223005, 2.26845101979555256413299697067, 2.38572041733964948988143809178, 2.38924989557373394682596906014, 2.45774456996433370157818986476, 2.50938415484935028289251819199, 2.61775868549872672952302147178, 2.87767050212256409319558424249, 3.26251046505719505598509328709, 3.29990016093481708768365530040, 3.38044982396807429428174909737, 3.44413110419457538944626556062, 3.54020780286309454517647484114
Plot not available for L-functions of degree greater than 10.
