| L(s) = 1 | − 8·11-s + 7·16-s − 8·41-s + 18·81-s − 52·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 56·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 2.41·11-s + 7/4·16-s − 1.24·41-s + 2·81-s − 4.72·121-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 + 0.0760·173-s − 4.22·176-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 + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9444849786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9444849786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 7 T^{4} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| good | 3 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | \( ( 1 + T + p T^{2} )^{8} \) |
| 13 | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | \( ( 1 - 529 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - p T^{2} )^{8} \) |
| 37 | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | \( ( 1 + T + p T^{2} )^{8} \) |
| 43 | \( ( 1 + 1778 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 1058 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4942 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 2471 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 9791 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 12791 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 9 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 2498 T^{4} + p^{4} T^{8} )^{2} \) |
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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
−5.72192331135833862938208238255, −5.42059046212967626140775437750, −5.29462071968932163047045300524, −5.23773910734911636484899946313, −5.08491451364155897386370354443, −4.95830896788192943739765852871, −4.94437029922387267978666672080, −4.62438911605500284573794078914, −4.58294297459241778671892292713, −3.96998802813972418392239807688, −3.96743892239597276226758213505, −3.86930602517432593438216884136, −3.82825159263916341201203205239, −3.53171594545448435277774914899, −3.35006689915916372868079898635, −3.02363864448979790835326015525, −2.79008691724919034083202752057, −2.73162574121744782582008007711, −2.55241550909841121118075188391, −2.42096631410509437341543282928, −2.04942555988121327520914368294, −1.59397794520837719506241506604, −1.48075950540778837325711794299, −1.15409946193346148089131271649, −0.38629434423270937956071929997,
0.38629434423270937956071929997, 1.15409946193346148089131271649, 1.48075950540778837325711794299, 1.59397794520837719506241506604, 2.04942555988121327520914368294, 2.42096631410509437341543282928, 2.55241550909841121118075188391, 2.73162574121744782582008007711, 2.79008691724919034083202752057, 3.02363864448979790835326015525, 3.35006689915916372868079898635, 3.53171594545448435277774914899, 3.82825159263916341201203205239, 3.86930602517432593438216884136, 3.96743892239597276226758213505, 3.96998802813972418392239807688, 4.58294297459241778671892292713, 4.62438911605500284573794078914, 4.94437029922387267978666672080, 4.95830896788192943739765852871, 5.08491451364155897386370354443, 5.23773910734911636484899946313, 5.29462071968932163047045300524, 5.42059046212967626140775437750, 5.72192331135833862938208238255
Plot not available for L-functions of degree greater than 10.
