| L(s) = 1 | − 8·9-s − 8·25-s − 16·37-s + 16·43-s + 16·67-s − 96·79-s + 32·81-s − 80·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 8/3·9-s − 8/5·25-s − 2.63·37-s + 2.43·43-s + 1.95·67-s − 10.8·79-s + 32/9·81-s − 7.66·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02143436938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02143436938\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 8 T^{2} + 32 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 4 T^{2} + 22 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 - 20 T^{2} + 310 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 32 T^{2} + 672 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 72 T^{2} + 2016 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 92 T^{2} + 3670 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 44 T^{2} + 2374 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 + 64 T^{2} + 3136 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 172 T^{2} + 11782 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 60 T^{2} + 5366 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 216 T^{2} + 18528 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 164 T^{2} + 14134 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 140 T^{2} + 12934 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 288 T^{2} + 31392 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 24 T + 294 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 216 T^{2} + 22080 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 192 T^{2} + 23136 T^{4} + 192 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 288 T^{2} + 37632 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.72302661867747688125357515035, −3.61796566189662466083855827085, −3.45174146177999038357779195623, −3.42392716048744615524545201041, −3.27702511599861213773802249449, −3.11408570326899965010574615101, −2.96719688861514156068211220254, −2.76361223644679611184172829631, −2.71030768692697089242155302033, −2.66843926847476948370943331272, −2.54383073550500684182081132216, −2.53587416094484935983975903442, −2.52473348585138582197259674456, −1.94997587399778394600161349384, −1.91599670978182431440401732830, −1.86365656853268167394993691815, −1.58205988397000464744294012979, −1.52320901332099071695158853064, −1.44501496177535091228482257395, −1.39972802345455683515505351104, −0.947153475658243567935198930440, −0.58906196831709040177937998865, −0.49275489396318029562227863247, −0.45358670175857144881950594923, −0.01816266038891747683578814577,
0.01816266038891747683578814577, 0.45358670175857144881950594923, 0.49275489396318029562227863247, 0.58906196831709040177937998865, 0.947153475658243567935198930440, 1.39972802345455683515505351104, 1.44501496177535091228482257395, 1.52320901332099071695158853064, 1.58205988397000464744294012979, 1.86365656853268167394993691815, 1.91599670978182431440401732830, 1.94997587399778394600161349384, 2.52473348585138582197259674456, 2.53587416094484935983975903442, 2.54383073550500684182081132216, 2.66843926847476948370943331272, 2.71030768692697089242155302033, 2.76361223644679611184172829631, 2.96719688861514156068211220254, 3.11408570326899965010574615101, 3.27702511599861213773802249449, 3.42392716048744615524545201041, 3.45174146177999038357779195623, 3.61796566189662466083855827085, 3.72302661867747688125357515035
Plot not available for L-functions of degree greater than 10.
