| L(s) = 1 | + 3·5-s + 6·17-s − 6·19-s + 30·23-s + 9·25-s − 16·31-s − 48·47-s + 137·49-s − 192·53-s + 38·61-s + 6·79-s + 288·83-s + 18·85-s − 18·95-s − 18·107-s − 226·109-s − 564·113-s + 90·115-s + 129·121-s + 102·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 48·155-s + ⋯ |
| L(s) = 1 | + 3/5·5-s + 6/17·17-s − 0.315·19-s + 1.30·23-s + 9/25·25-s − 0.516·31-s − 1.02·47-s + 2.79·49-s − 3.62·53-s + 0.622·61-s + 6/79·79-s + 3.46·83-s + 0.211·85-s − 0.189·95-s − 0.168·107-s − 2.07·109-s − 4.99·113-s + 0.782·115-s + 1.06·121-s + 0.815·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 0.309·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.558189264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.558189264\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 3 T - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 137 T^{2} + 9024 T^{4} - 137 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 129 T^{2} + 10400 T^{4} - 129 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 136 T^{2} - 5970 T^{4} - 136 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 3 T + 528 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 3 T + 254 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 15 T + 1062 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 57 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 1522 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3186 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 776 T^{2} + 5716590 T^{4} - 776 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 24 T + 3726 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 96 T + 6041 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 7044 T^{2} + 28934630 T^{4} - 7044 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 19 T + 7062 T^{2} - 19 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6584 T^{2} + 19350606 T^{4} - 6584 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 3208 T^{2} + 50078094 T^{4} - 3208 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 9521 T^{2} + 50769360 T^{4} - 9521 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 3 T + 8252 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 144 T + 13737 T^{2} - 144 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6984 T^{2} + 4586510 T^{4} - 6984 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 1879 T^{2} + 54186000 T^{4} + 1879 p^{4} T^{6} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.25463217206279760292280676176, −6.07213391360948891186447392939, −5.92154072264997632557879587901, −5.49380056410662025556531619293, −5.33266981132248607014922667984, −5.28446453240214151900805790271, −5.12956633168430795171826042771, −4.77196822507263347157883499006, −4.55443008358671759128392437993, −4.46434148931892996586207064854, −3.99880228810112708617311903516, −3.90534232403851293415952699847, −3.66835289118221716613950145951, −3.33450483032914835040803969372, −3.09713444608772066687689511379, −3.04831472959587823557731024274, −2.45767643306983300593775718562, −2.37979568795128069548292038516, −2.37766732125980999550645154237, −1.65543679345467919426987487368, −1.48737997847937688239799373727, −1.47463515624593381083772208146, −0.835810586421629522373645965072, −0.68546635599078927958049217665, −0.18879155904879151652914180847,
0.18879155904879151652914180847, 0.68546635599078927958049217665, 0.835810586421629522373645965072, 1.47463515624593381083772208146, 1.48737997847937688239799373727, 1.65543679345467919426987487368, 2.37766732125980999550645154237, 2.37979568795128069548292038516, 2.45767643306983300593775718562, 3.04831472959587823557731024274, 3.09713444608772066687689511379, 3.33450483032914835040803969372, 3.66835289118221716613950145951, 3.90534232403851293415952699847, 3.99880228810112708617311903516, 4.46434148931892996586207064854, 4.55443008358671759128392437993, 4.77196822507263347157883499006, 5.12956633168430795171826042771, 5.28446453240214151900805790271, 5.33266981132248607014922667984, 5.49380056410662025556531619293, 5.92154072264997632557879587901, 6.07213391360948891186447392939, 6.25463217206279760292280676176
