| L(s) = 1 | + 12·9-s − 10·16-s − 56·49-s + 90·81-s + 128·103-s + 127-s + 131-s + 137-s + 139-s − 120·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 4·9-s − 5/2·16-s − 8·49-s + 10·81-s + 12.6·103-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-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 + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.839165855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.839165855\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 - p T^{2} )^{4} \) |
| 11 | \( 1 - 190 T^{4} + p^{4} T^{8} \) |
| 13 | \( ( 1 + p T^{2} )^{4} \) |
| good | 2 | \( ( 1 + 5 T^{4} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 + 2 T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + p T^{2} )^{8} \) |
| 17 | \( ( 1 + p T^{2} )^{8} \) |
| 19 | \( ( 1 + p T^{2} )^{8} \) |
| 23 | \( ( 1 - p T^{2} )^{8} \) |
| 29 | \( ( 1 + p T^{2} )^{8} \) |
| 31 | \( ( 1 - p T^{2} )^{8} \) |
| 37 | \( ( 1 - p T^{2} )^{8} \) |
| 41 | \( ( 1 + 2930 T^{4} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 4 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 47 | \( ( 1 + 4370 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - p T^{2} )^{8} \) |
| 59 | \( ( 1 - 6910 T^{4} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - p T^{2} )^{8} \) |
| 71 | \( ( 1 - 3790 T^{4} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + p T^{2} )^{8} \) |
| 79 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 13730 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 9550 T^{4} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - p T^{2} )^{8} \) |
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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
−4.80954684346471203799228882454, −4.72442412879830577960760216072, −4.67620890011258919702863290808, −4.56249832888358429430801712619, −4.51338293167750547674066886430, −4.46999776006899993532603180947, −4.11937665389295349191829403198, −3.84254860659972554947160598838, −3.61569968096864958134030206640, −3.61438339118209003024604374882, −3.59660616470243075692976401015, −3.35953776803622668256751741464, −3.15086478670018742500236617256, −3.13991165025232550553253554097, −2.75735579738725682876418946109, −2.48654163936732271801473906890, −2.13151846616453730628057715606, −2.07959084834684662519247795510, −2.02072878132481459457224833921, −1.85444885260487459481646255009, −1.71130098612822103426700722198, −1.30477914861469934398728760290, −1.12874593625313832196527391657, −0.918782567530916970510357489167, −0.23725061969688850526275835641,
0.23725061969688850526275835641, 0.918782567530916970510357489167, 1.12874593625313832196527391657, 1.30477914861469934398728760290, 1.71130098612822103426700722198, 1.85444885260487459481646255009, 2.02072878132481459457224833921, 2.07959084834684662519247795510, 2.13151846616453730628057715606, 2.48654163936732271801473906890, 2.75735579738725682876418946109, 3.13991165025232550553253554097, 3.15086478670018742500236617256, 3.35953776803622668256751741464, 3.59660616470243075692976401015, 3.61438339118209003024604374882, 3.61569968096864958134030206640, 3.84254860659972554947160598838, 4.11937665389295349191829403198, 4.46999776006899993532603180947, 4.51338293167750547674066886430, 4.56249832888358429430801712619, 4.67620890011258919702863290808, 4.72442412879830577960760216072, 4.80954684346471203799228882454
Plot not available for L-functions of degree greater than 10.
