| L(s) = 1 | + 8·5-s + 24·13-s + 32·25-s − 136·29-s − 112·37-s + 116·41-s + 224·61-s + 192·65-s − 292·73-s + 18·81-s − 160·89-s + 28·97-s + 280·101-s − 56·109-s − 532·113-s + 264·125-s + ⋯ |
| L(s) = 1 | + 8/5·5-s + 1.84·13-s + 1.27·25-s − 4.68·29-s − 3.02·37-s + 2.82·41-s + 3.67·61-s + 2.95·65-s − 4·73-s + 2/9·81-s − 1.79·89-s + 0.288·97-s + 2.77·101-s − 0.513·109-s − 4.70·113-s + 2.11·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.486095605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.486095605\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 - 2 p^{2} T^{4} + p^{8} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - 4318 T^{4} + p^{8} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 15662 T^{4} + p^{8} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 238 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 68 T + 2312 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 + 1400162 T^{4} + p^{8} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 56 T + 1568 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 58 T + 1682 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 758 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 3458 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 3502 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 4022 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 112 T + 6272 T^{2} - 112 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 8918 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - 17622718 T^{4} + p^{8} T^{8} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{4}( 1 + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^3$ | \( 1 - 54879358 T^{4} + p^{8} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 10838 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 144 T + p^{2} T^{2} )^{2}( 1 + 130 T + p^{2} T^{2} )^{2} \) |
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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
−8.508225533316584570593533699707, −8.305326481108852068175706616988, −7.940029984186449204328186944290, −7.53133233109347274243144788852, −7.40918747654824948789106190054, −7.14858195288422254035688980373, −6.75480678450467999974283674502, −6.70753925706633279258362407594, −6.20153440761128225859758326876, −5.86209390863254027440011598086, −5.68445793360125041157822324765, −5.64454678347891779938922239060, −5.28115708650358167416895993556, −5.22919814783445074014845027663, −4.37255761783161996343545259687, −4.19535387569604683892121626515, −3.95997232469503528107305996298, −3.45714448970742161048162218270, −3.40956538084801295214020536267, −2.80976995145413241138283542614, −2.31481989912461104146744014600, −1.81191202592462716395634374690, −1.78283540238824155629699126260, −1.26539515925207408943989628701, −0.44455312113385041902181406550,
0.44455312113385041902181406550, 1.26539515925207408943989628701, 1.78283540238824155629699126260, 1.81191202592462716395634374690, 2.31481989912461104146744014600, 2.80976995145413241138283542614, 3.40956538084801295214020536267, 3.45714448970742161048162218270, 3.95997232469503528107305996298, 4.19535387569604683892121626515, 4.37255761783161996343545259687, 5.22919814783445074014845027663, 5.28115708650358167416895993556, 5.64454678347891779938922239060, 5.68445793360125041157822324765, 5.86209390863254027440011598086, 6.20153440761128225859758326876, 6.70753925706633279258362407594, 6.75480678450467999974283674502, 7.14858195288422254035688980373, 7.40918747654824948789106190054, 7.53133233109347274243144788852, 7.940029984186449204328186944290, 8.305326481108852068175706616988, 8.508225533316584570593533699707
