| L(s) = 1 | + 4-s + 4·7-s − 13-s + 16-s + 6·19-s − 8·25-s + 4·28-s + 8·31-s + 2·37-s + 5·43-s + 9·49-s − 52-s + 13·61-s + 64-s + 13·67-s + 6·73-s + 6·76-s − 26·79-s − 4·91-s − 9·97-s − 8·100-s + 103-s + 6·109-s + 4·112-s + 16·121-s + 8·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.51·7-s − 0.277·13-s + 1/4·16-s + 1.37·19-s − 8/5·25-s + 0.755·28-s + 1.43·31-s + 0.328·37-s + 0.762·43-s + 9/7·49-s − 0.138·52-s + 1.66·61-s + 1/8·64-s + 1.58·67-s + 0.702·73-s + 0.688·76-s − 2.92·79-s − 0.419·91-s − 0.913·97-s − 4/5·100-s + 0.0985·103-s + 0.574·109-s + 0.377·112-s + 1.45·121-s + 0.718·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.272973904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.272973904\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.010175285921456798918495035833, −7.49790127876404696876795918612, −7.26929058881614633896431831911, −6.82014086489090632191839873114, −6.14946629248884233622015749706, −5.71897637147201810863010022640, −5.40776167026155709609566990883, −4.86360719312348575597517207169, −4.42816347844486775133422540223, −3.92746783558925231503567404719, −3.33734387614801664881659815949, −2.60327123881347549029197899208, −2.19944632133571925165013486727, −1.50017402387809065022404247946, −0.852682508661966392049472693168,
0.852682508661966392049472693168, 1.50017402387809065022404247946, 2.19944632133571925165013486727, 2.60327123881347549029197899208, 3.33734387614801664881659815949, 3.92746783558925231503567404719, 4.42816347844486775133422540223, 4.86360719312348575597517207169, 5.40776167026155709609566990883, 5.71897637147201810863010022640, 6.14946629248884233622015749706, 6.82014086489090632191839873114, 7.26929058881614633896431831911, 7.49790127876404696876795918612, 8.010175285921456798918495035833
