| L(s) = 1 | − 8·4-s + 98·7-s + 1.26e3·19-s − 1.24e3·25-s − 784·28-s + 3.70e3·31-s + 2.08e3·37-s + 1.16e4·43-s + 2.40e3·49-s − 6.01e3·61-s + 512·64-s + 3.03e3·67-s + 5.50e3·73-s − 1.01e4·76-s − 7.27e3·79-s + 9.95e3·100-s + 2.30e4·103-s − 7.14e3·109-s + 2.34e4·121-s − 2.96e4·124-s + 127-s + 131-s + 1.24e5·133-s + 137-s + 139-s − 1.66e4·148-s + 149-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2·7-s + 3.50·19-s − 1.99·25-s − 28-s + 3.85·31-s + 1.51·37-s + 6.31·43-s + 49-s − 1.61·61-s + 1/8·64-s + 0.675·67-s + 1.03·73-s − 1.75·76-s − 1.16·79-s + 0.995·100-s + 2.16·103-s − 0.601·109-s + 1.60·121-s − 1.92·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 7.01·133-s + 5.32e−5·137-s + 5.17e−5·139-s − 0.759·148-s + 4.50e−5·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(14.57028704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.57028704\) |
| \(L(3)\) |
|
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 | $C_2^2$ | \( 1 + p^{3} T^{2} + p^{6} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 + 1244 T^{2} + 1156911 T^{4} + 1244 p^{8} T^{6} + p^{16} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 23450 T^{2} + 335543619 T^{4} - 23450 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 32822 T^{2} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 110588 T^{2} + 5253948303 T^{4} + 110588 p^{8} T^{6} + p^{16} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 633 T + 263884 T^{2} - 633 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 519920 T^{2} + 192005821119 T^{4} - 519920 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 957280 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1851 T + 2065588 T^{2} - 1851 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 1040 T - 792561 T^{2} - 1040 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 4060172 T^{2} + p^{8} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2917 T + p^{4} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 1950292 T^{2} - 20007647776497 T^{4} - 1950292 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 12299056 T^{2} + 89007088079775 T^{4} + 12299056 p^{8} T^{6} + p^{16} T^{8} \) |
| 59 | $C_2^3$ | \( 1 + 21141578 T^{2} + 300135882725763 T^{4} + 21141578 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 3009 T + 16863868 T^{2} + 3009 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 1516 T - 17852865 T^{2} - 1516 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 45378362 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 2751 T + 30920908 T^{2} - 2751 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 3638 T - 25715037 T^{2} + 3638 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 35936758 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 124613516 T^{2} + 11591939564180175 T^{4} + 124613516 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 71029487 T^{2} + p^{8} T^{4} )^{2} \) |
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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
−7.75519040136726880327696735171, −7.55729448689915869821765185960, −7.18186523729477192422337898251, −7.00965176728261603645017880385, −6.36026274745821738915154506999, −6.33879684877450685056700702259, −5.96795383929485976925195149961, −5.57159626539020944364849661565, −5.52776757623350949236459018722, −5.43853057246473657679976044871, −4.88698230704296651332981548723, −4.53171519098750740238014228093, −4.40660834296644446998265084362, −4.25862907883297663975329847358, −4.12727789922776761739238418131, −3.41422836781624146197151879830, −3.12790561997872419075453990584, −2.88322922223632200974098277360, −2.51854853745472622559187381517, −2.17968108778566805882698836672, −1.81640466261162295090901021372, −1.20811395704443418158831209310, −1.02837888653844985545558635342, −0.69379947080997464471942229656, −0.64144758669847398456222481743,
0.64144758669847398456222481743, 0.69379947080997464471942229656, 1.02837888653844985545558635342, 1.20811395704443418158831209310, 1.81640466261162295090901021372, 2.17968108778566805882698836672, 2.51854853745472622559187381517, 2.88322922223632200974098277360, 3.12790561997872419075453990584, 3.41422836781624146197151879830, 4.12727789922776761739238418131, 4.25862907883297663975329847358, 4.40660834296644446998265084362, 4.53171519098750740238014228093, 4.88698230704296651332981548723, 5.43853057246473657679976044871, 5.52776757623350949236459018722, 5.57159626539020944364849661565, 5.96795383929485976925195149961, 6.33879684877450685056700702259, 6.36026274745821738915154506999, 7.00965176728261603645017880385, 7.18186523729477192422337898251, 7.55729448689915869821765185960, 7.75519040136726880327696735171
