| L(s) = 1 | − 2·3-s + 2·9-s + 4·11-s − 4·13-s − 12·17-s + 6·19-s − 6·27-s − 2·29-s − 20·31-s − 8·33-s + 6·37-s + 8·39-s − 12·43-s − 4·47-s − 49-s + 24·51-s + 18·53-s − 12·57-s + 2·59-s + 16·67-s − 8·79-s + 11·81-s + 10·83-s + 4·87-s + 40·93-s − 4·97-s + 8·99-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 2/3·9-s + 1.20·11-s − 1.10·13-s − 2.91·17-s + 1.37·19-s − 1.15·27-s − 0.371·29-s − 3.59·31-s − 1.39·33-s + 0.986·37-s + 1.28·39-s − 1.82·43-s − 0.583·47-s − 1/7·49-s + 3.36·51-s + 2.47·53-s − 1.58·57-s + 0.260·59-s + 1.95·67-s − 0.900·79-s + 11/9·81-s + 1.09·83-s + 0.428·87-s + 4.14·93-s − 0.406·97-s + 0.804·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12845056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12845056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 174 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.440287009060625157677459105039, −7.917938796026827759911746443961, −7.46210067843467495822047838550, −7.05830537809629821034437017401, −6.79396448796361483714131232033, −6.77535871634740086493230651392, −6.08267192462224192712747291132, −5.70277842224657234115772809869, −5.33238298363205280371230262034, −5.06293670951639563209588926909, −4.66468158522138578700967490351, −4.00746153818772332526731938225, −3.88288141314023248493577835272, −3.47070478609745531333952863288, −2.58257068498875587442373205522, −2.14490216840986210925220828057, −1.78945735514046629016215899328, −1.09082795927623013946800225721, 0, 0,
1.09082795927623013946800225721, 1.78945735514046629016215899328, 2.14490216840986210925220828057, 2.58257068498875587442373205522, 3.47070478609745531333952863288, 3.88288141314023248493577835272, 4.00746153818772332526731938225, 4.66468158522138578700967490351, 5.06293670951639563209588926909, 5.33238298363205280371230262034, 5.70277842224657234115772809869, 6.08267192462224192712747291132, 6.77535871634740086493230651392, 6.79396448796361483714131232033, 7.05830537809629821034437017401, 7.46210067843467495822047838550, 7.917938796026827759911746443961, 8.440287009060625157677459105039
