| L(s) = 1 | − 48·41-s − 16·43-s + 16·49-s + 48·67-s + 16·83-s + 16·89-s − 32·107-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 64·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 7.49·41-s − 2.43·43-s + 16/7·49-s + 5.86·67-s + 1.75·83-s + 1.69·89-s − 3.09·107-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 − 4.92·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4430514355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4430514355\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| good | 7 | \( ( 1 - 8 T^{2} + 30 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 + 32 T^{2} + 510 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 24 T^{2} + 638 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - p T^{2} )^{8} \) |
| 29 | \( ( 1 - 48 T^{2} + 1502 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 80 T^{2} + 3582 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 6 T + p T^{2} )^{8} \) |
| 43 | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 144 T^{2} + 10046 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 176 T^{2} + 13950 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 4 T + 86 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 97 | \( ( 1 - 212 T^{2} + 28710 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.59667259589782566899619244285, −3.56523692950135190375610911617, −3.44967109021392019549991321715, −3.44067254180529420341981599715, −3.18218742723311018042917160468, −3.11952704569113142957106461583, −3.08965076835682512648064748049, −2.67737363781925643105927537522, −2.60850491491152835356521165892, −2.58341382706289013638645321571, −2.30968050079228806184489651213, −2.29894247864117417237016901876, −2.24039057789340731232075460272, −2.06259382541611049206974723662, −1.89784122970184429528566597000, −1.59347984446830671147871182914, −1.58808052697300086804185676315, −1.57039487008651901158980881101, −1.47042404211100835829295646285, −0.992494177116205024785220714603, −0.957336320446411597133264084955, −0.925013063043467481159694749146, −0.43584334449993357908251174287, −0.38849009286953441849852485164, −0.06586304157960416071207226202,
0.06586304157960416071207226202, 0.38849009286953441849852485164, 0.43584334449993357908251174287, 0.925013063043467481159694749146, 0.957336320446411597133264084955, 0.992494177116205024785220714603, 1.47042404211100835829295646285, 1.57039487008651901158980881101, 1.58808052697300086804185676315, 1.59347984446830671147871182914, 1.89784122970184429528566597000, 2.06259382541611049206974723662, 2.24039057789340731232075460272, 2.29894247864117417237016901876, 2.30968050079228806184489651213, 2.58341382706289013638645321571, 2.60850491491152835356521165892, 2.67737363781925643105927537522, 3.08965076835682512648064748049, 3.11952704569113142957106461583, 3.18218742723311018042917160468, 3.44067254180529420341981599715, 3.44967109021392019549991321715, 3.56523692950135190375610911617, 3.59667259589782566899619244285
Plot not available for L-functions of degree greater than 10.
