| L(s) = 1 | + 14·7-s + 66·19-s − 10·25-s − 70·31-s + 14·37-s + 30·43-s − 49-s + 66·61-s + 74·67-s + 210·73-s + 212·79-s − 202·97-s − 110·103-s − 304·109-s + 467·121-s + 127-s + 131-s + 924·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 386·169-s + 173-s + ⋯ |
| L(s) = 1 | + 2·7-s + 3.47·19-s − 2/5·25-s − 2.25·31-s + 0.378·37-s + 0.697·43-s − 0.0204·49-s + 1.08·61-s + 1.10·67-s + 2.87·73-s + 2.68·79-s − 2.08·97-s − 1.06·103-s − 2.78·109-s + 3.85·121-s + 0.00787·127-s + 0.00763·131-s + 6.94·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.28·169-s + 0.00578·173-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(11.29210183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.29210183\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 7 | $D_{4}$ | \( ( 1 - p T + 74 T^{2} - p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 467 T^{2} + 83768 T^{4} - 467 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 193 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 383 T^{2} + 159308 T^{4} - 383 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 33 T + 958 T^{2} - 33 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 163 T^{2} + 328488 T^{4} - 163 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 611 T^{2} + 766616 T^{4} - 611 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 35 T + 2192 T^{2} + 35 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 7 T + 2424 T^{2} - 7 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2012 T^{2} + 2951558 T^{4} - 2012 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 15 T - 632 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 5519 T^{2} + 14632796 T^{4} - 5519 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 10844 T^{2} + 45141926 T^{4} - 10844 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4064 T^{2} + 16169246 T^{4} - 4064 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 33 T + 6808 T^{2} - 33 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 37 T - 1156 T^{2} - 37 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12896 T^{2} + 90645566 T^{4} - 12896 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 105 T + 11638 T^{2} - 105 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 106 T + 12971 T^{2} - 106 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 3004 T^{2} - 11069274 T^{4} - 3004 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 17572 T^{2} + 154570758 T^{4} - 17572 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 101 T + 13212 T^{2} + 101 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.26662603223151319316138488716, −6.00990384671469887284051916572, −5.72979646010355007786866405711, −5.38580423820887194992748028340, −5.32464698764476557374873951066, −5.28442780349641810507294417325, −5.11601456567824963588161275092, −4.87288773743305308935107962791, −4.69736568887619909430579130337, −4.35855838125884420859602697748, −3.99075766282014169466406777993, −3.87668710384911953550714422982, −3.64051735176215501243804471911, −3.50977833841990972039644604050, −3.20878889192784550434249984733, −2.92749037495528434493485117948, −2.60141938621675116133282806183, −2.25845525206748658559377864071, −2.20967773048136790012895006137, −1.70107597878167829606537642695, −1.56483334654043674192434251460, −1.22221813633950190756975838550, −1.10538003168572306237084706027, −0.50528694411791531396080150703, −0.47524965928426993756601391137,
0.47524965928426993756601391137, 0.50528694411791531396080150703, 1.10538003168572306237084706027, 1.22221813633950190756975838550, 1.56483334654043674192434251460, 1.70107597878167829606537642695, 2.20967773048136790012895006137, 2.25845525206748658559377864071, 2.60141938621675116133282806183, 2.92749037495528434493485117948, 3.20878889192784550434249984733, 3.50977833841990972039644604050, 3.64051735176215501243804471911, 3.87668710384911953550714422982, 3.99075766282014169466406777993, 4.35855838125884420859602697748, 4.69736568887619909430579130337, 4.87288773743305308935107962791, 5.11601456567824963588161275092, 5.28442780349641810507294417325, 5.32464698764476557374873951066, 5.38580423820887194992748028340, 5.72979646010355007786866405711, 6.00990384671469887284051916572, 6.26662603223151319316138488716
