| L(s) = 1 | − 3-s + 6·5-s − 7-s − 6·11-s − 6·15-s − 12·17-s − 7·19-s + 21-s + 19·25-s + 27-s + 5·31-s + 6·33-s − 6·35-s + 37-s − 6·47-s − 6·49-s + 12·51-s − 36·55-s + 7·57-s + 3·67-s + 15·73-s − 19·75-s + 6·77-s + 27·79-s − 81-s − 12·83-s − 72·85-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 2.68·5-s − 0.377·7-s − 1.80·11-s − 1.54·15-s − 2.91·17-s − 1.60·19-s + 0.218·21-s + 19/5·25-s + 0.192·27-s + 0.898·31-s + 1.04·33-s − 1.01·35-s + 0.164·37-s − 0.875·47-s − 6/7·49-s + 1.68·51-s − 4.85·55-s + 0.927·57-s + 0.366·67-s + 1.75·73-s − 2.19·75-s + 0.683·77-s + 3.03·79-s − 1/9·81-s − 1.31·83-s − 7.80·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.372702478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.372702478\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 27 T + 322 T^{2} - 27 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−10.01358578187678467973285965681, −9.401125670394863516360453610713, −9.138583539104255913167845357693, −8.774647574478406259152580327013, −8.111294598068141072307414067493, −8.104652258271756420031806969559, −7.06514296176328701959695351664, −6.58943109740842429512723640410, −6.57688822912740895268988219936, −5.98713988355744226536585042432, −5.92557343419238126302057919266, −5.16019223796695592025845711692, −4.78941484943809288376211152268, −4.72163424407434968305944362402, −3.80968836216855497602501613923, −2.90449079513734976829214785036, −2.30824904945924204827088617707, −2.27417730767535000593169233888, −1.76067376641368651580485753200, −0.45446641738593379963099074030,
0.45446641738593379963099074030, 1.76067376641368651580485753200, 2.27417730767535000593169233888, 2.30824904945924204827088617707, 2.90449079513734976829214785036, 3.80968836216855497602501613923, 4.72163424407434968305944362402, 4.78941484943809288376211152268, 5.16019223796695592025845711692, 5.92557343419238126302057919266, 5.98713988355744226536585042432, 6.57688822912740895268988219936, 6.58943109740842429512723640410, 7.06514296176328701959695351664, 8.104652258271756420031806969559, 8.111294598068141072307414067493, 8.774647574478406259152580327013, 9.138583539104255913167845357693, 9.401125670394863516360453610713, 10.01358578187678467973285965681
