| L(s) = 1 | − 3-s + 2·7-s + 9-s + 4·13-s + 8·19-s − 2·21-s − 2·25-s − 27-s + 2·31-s + 8·37-s − 4·39-s + 12·43-s − 10·49-s − 8·57-s − 8·61-s + 2·63-s + 4·67-s + 12·73-s + 2·75-s + 18·79-s + 81-s + 8·91-s − 2·93-s − 20·97-s + 14·103-s − 4·109-s − 8·111-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.10·13-s + 1.83·19-s − 0.436·21-s − 2/5·25-s − 0.192·27-s + 0.359·31-s + 1.31·37-s − 0.640·39-s + 1.82·43-s − 1.42·49-s − 1.05·57-s − 1.02·61-s + 0.251·63-s + 0.488·67-s + 1.40·73-s + 0.230·75-s + 2.02·79-s + 1/9·81-s + 0.838·91-s − 0.207·93-s − 2.03·97-s + 1.37·103-s − 0.383·109-s − 0.759·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.001801645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.001801645\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.454473814331367329134444556729, −8.051017571788834912747542741922, −7.68334146859806638809398916303, −7.38058155411024317978703466459, −6.58715794575130066448620490328, −6.27141309034599809760146017112, −5.80383797224955526286135852540, −5.21261506766993869393555765470, −4.95445769558805817996775944144, −4.22361443043057482993898802130, −3.78312884947693409549778140192, −3.12280025830169798817173767961, −2.39398825782297191987730281665, −1.45279728603284071454293255441, −0.902374709471473533403127085790,
0.902374709471473533403127085790, 1.45279728603284071454293255441, 2.39398825782297191987730281665, 3.12280025830169798817173767961, 3.78312884947693409549778140192, 4.22361443043057482993898802130, 4.95445769558805817996775944144, 5.21261506766993869393555765470, 5.80383797224955526286135852540, 6.27141309034599809760146017112, 6.58715794575130066448620490328, 7.38058155411024317978703466459, 7.68334146859806638809398916303, 8.051017571788834912747542741922, 8.454473814331367329134444556729
