| L(s) = 1 | − 2·3-s + 3·4-s + 9-s − 6·12-s + 5·16-s − 12·17-s + 6·23-s − 6·25-s + 4·27-s + 6·29-s + 3·36-s + 6·43-s − 10·48-s + 6·49-s + 24·51-s + 12·53-s + 3·64-s − 36·68-s − 12·69-s + 12·75-s + 18·79-s − 11·81-s − 12·87-s + 18·92-s − 18·100-s − 6·101-s + 18·103-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 3/2·4-s + 1/3·9-s − 1.73·12-s + 5/4·16-s − 2.91·17-s + 1.25·23-s − 6/5·25-s + 0.769·27-s + 1.11·29-s + 1/2·36-s + 0.914·43-s − 1.44·48-s + 6/7·49-s + 3.36·51-s + 1.64·53-s + 3/8·64-s − 4.36·68-s − 1.44·69-s + 1.38·75-s + 2.02·79-s − 1.22·81-s − 1.28·87-s + 1.87·92-s − 9/5·100-s − 0.597·101-s + 1.77·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.408580889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.408580889\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 108 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
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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
−8.970125848173998994684760506018, −8.487584631003476174134158097609, −7.87642045358338714214931539289, −7.15590525799677180463941085958, −6.99318637816637186538739817397, −6.47935424681747775730110650578, −6.22812548115877397261474576184, −5.67922882109248776419048107168, −5.10139148210534141637774739016, −4.50628813901430351682822026380, −4.03352559081267628286098427022, −3.03544178085600623555214246551, −2.40585565231048627456387454957, −1.98383303527603389108299202363, −0.73646744675659069181676212991,
0.73646744675659069181676212991, 1.98383303527603389108299202363, 2.40585565231048627456387454957, 3.03544178085600623555214246551, 4.03352559081267628286098427022, 4.50628813901430351682822026380, 5.10139148210534141637774739016, 5.67922882109248776419048107168, 6.22812548115877397261474576184, 6.47935424681747775730110650578, 6.99318637816637186538739817397, 7.15590525799677180463941085958, 7.87642045358338714214931539289, 8.487584631003476174134158097609, 8.970125848173998994684760506018
