| L(s) = 1 | − 4·9-s − 4·25-s + 16·29-s − 16·37-s + 18·49-s − 16·53-s + 8·81-s + 24·109-s + 16·113-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 88·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 4/3·9-s − 4/5·25-s + 2.97·29-s − 2.63·37-s + 18/7·49-s − 2.19·53-s + 8/9·81-s + 2.29·109-s + 1.50·113-s + 2.90·121-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 + 6.76·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8024145337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8024145337\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T^{2} )^{4} \) |
| 7 | \( 1 - 18 T^{2} + 162 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 3 | \( ( 1 + 2 T^{2} + 2 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 16 T^{2} + 238 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 16 T^{2} + 30 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 48 T^{2} + 1230 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 18 T^{2} + 1122 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 31 | \( ( 1 + 68 T^{2} + 2806 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 - 112 T^{2} + 5886 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 114 T^{2} + 6114 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 62 T^{2} + 4002 T^{4} + 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 59 | \( ( 1 + 112 T^{2} + 6766 T^{4} + 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 50 T^{2} + 610 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 192 T^{2} + 19230 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 32 T^{2} + 5406 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 296 T^{2} + 34318 T^{4} - 296 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 322 T^{2} + 39682 T^{4} + 322 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 336 T^{2} + 46430 T^{4} - 336 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.84729724808249705906702477725, −3.80363489606849949036209430998, −3.47272421971942313459534166513, −3.25449383255933261120266531512, −3.18206314759799960520684925133, −3.15611737980817596938762478897, −3.13088492225345527491572461518, −2.96565834590528914599729150789, −2.87323268773115603182658723468, −2.77817338383349115632459562878, −2.40938064840738184691841860154, −2.38807377598615793810661406788, −2.25799575551080620138605966327, −2.06833822270701356252698972111, −2.05510125885666237789316120808, −1.91930272147753124557716970056, −1.82986736809007935956284936199, −1.46573518066035793925626668171, −1.29961981988691068308088171071, −1.06543700321972732572875287819, −1.02782768039119822552053321600, −0.953113451549606412450749670396, −0.65026121636732610196920183241, −0.22512145870729395016356377901, −0.13944375179634199280380420981,
0.13944375179634199280380420981, 0.22512145870729395016356377901, 0.65026121636732610196920183241, 0.953113451549606412450749670396, 1.02782768039119822552053321600, 1.06543700321972732572875287819, 1.29961981988691068308088171071, 1.46573518066035793925626668171, 1.82986736809007935956284936199, 1.91930272147753124557716970056, 2.05510125885666237789316120808, 2.06833822270701356252698972111, 2.25799575551080620138605966327, 2.38807377598615793810661406788, 2.40938064840738184691841860154, 2.77817338383349115632459562878, 2.87323268773115603182658723468, 2.96565834590528914599729150789, 3.13088492225345527491572461518, 3.15611737980817596938762478897, 3.18206314759799960520684925133, 3.25449383255933261120266531512, 3.47272421971942313459534166513, 3.80363489606849949036209430998, 3.84729724808249705906702477725
Plot not available for L-functions of degree greater than 10.
