| L(s) = 1 | + 8·3-s + 36·9-s + 120·27-s + 48·41-s + 16·43-s + 16·49-s − 48·67-s + 330·81-s + 16·83-s − 16·89-s − 32·107-s + 384·123-s + 127-s + 128·129-s + 131-s + 137-s + 139-s + 128·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 64·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 4.61·3-s + 12·9-s + 23.0·27-s + 7.49·41-s + 2.43·43-s + 16/7·49-s − 5.86·67-s + 36.6·81-s + 1.75·83-s − 1.69·89-s − 3.09·107-s + 34.6·123-s + 0.0887·127-s + 11.2·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.5·147-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(87.61515205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(87.61515205\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{8} \) |
| 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
−4.14252665933310965606549318476, −4.13604111549093705642502881919, −3.97876483433869045877745799234, −3.97271593544276310604733644549, −3.77751093645530228103251386610, −3.46831383338552390302390837986, −3.44493382181137262083271094993, −3.39804892071928111134781678818, −3.35911607139303819623519933003, −2.98829563433511250980303596479, −2.66957267631923712674981011781, −2.60590676949872702473979671943, −2.55132825045072329110727062228, −2.53939693853637857921225167605, −2.51053011147161541989957919108, −2.47853034376631774370746372174, −2.41589874391804421472318883664, −1.85290777608434368963523476933, −1.62372681192832743913910652782, −1.58758779731163356425889584686, −1.41734781560005035492256066339, −1.15640268350797813103677450764, −1.11543588568303041514212141620, −0.61288261987233158384582365943, −0.57734720868758833853981147980,
0.57734720868758833853981147980, 0.61288261987233158384582365943, 1.11543588568303041514212141620, 1.15640268350797813103677450764, 1.41734781560005035492256066339, 1.58758779731163356425889584686, 1.62372681192832743913910652782, 1.85290777608434368963523476933, 2.41589874391804421472318883664, 2.47853034376631774370746372174, 2.51053011147161541989957919108, 2.53939693853637857921225167605, 2.55132825045072329110727062228, 2.60590676949872702473979671943, 2.66957267631923712674981011781, 2.98829563433511250980303596479, 3.35911607139303819623519933003, 3.39804892071928111134781678818, 3.44493382181137262083271094993, 3.46831383338552390302390837986, 3.77751093645530228103251386610, 3.97271593544276310604733644549, 3.97876483433869045877745799234, 4.13604111549093705642502881919, 4.14252665933310965606549318476
Plot not available for L-functions of degree greater than 10.
