| L(s) = 1 | − 2-s + 3-s − 6-s − 8·7-s + 8-s + 3·9-s + 6·11-s + 2·13-s + 8·14-s − 16-s + 6·17-s − 3·18-s − 8·21-s − 6·22-s + 6·23-s + 24-s + 5·25-s − 2·26-s + 8·27-s − 4·31-s + 6·33-s − 6·34-s + 20·37-s + 2·39-s + 9·41-s + 8·42-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.408·6-s − 3.02·7-s + 0.353·8-s + 9-s + 1.80·11-s + 0.554·13-s + 2.13·14-s − 1/4·16-s + 1.45·17-s − 0.707·18-s − 1.74·21-s − 1.27·22-s + 1.25·23-s + 0.204·24-s + 25-s − 0.392·26-s + 1.53·27-s − 0.718·31-s + 1.04·33-s − 1.02·34-s + 3.28·37-s + 0.320·39-s + 1.40·41-s + 1.23·42-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 521284 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 521284 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.618266836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.618266836\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−10.22062006650143921146447576915, −10.15761349996729779102090249759, −9.568211410120342513242644577001, −9.247545614676605058209101754583, −9.094741872857401152364938958964, −8.934927884322279446399086143565, −8.005691707083077555964883468116, −7.52892217139477526881306715143, −7.18781469341429282191450835697, −6.63058751192366131405794879734, −6.30792894084654795247437044598, −6.14010503213585919367476995879, −5.34119516713738043799628797589, −4.39614871962962577023010062234, −3.96951993316930154057935375907, −3.60308736951225359916574630362, −2.83705727483024362035365225448, −2.79886206316725918712261368610, −1.14546742610286588241836353242, −0.944022741958428169397854814197,
0.944022741958428169397854814197, 1.14546742610286588241836353242, 2.79886206316725918712261368610, 2.83705727483024362035365225448, 3.60308736951225359916574630362, 3.96951993316930154057935375907, 4.39614871962962577023010062234, 5.34119516713738043799628797589, 6.14010503213585919367476995879, 6.30792894084654795247437044598, 6.63058751192366131405794879734, 7.18781469341429282191450835697, 7.52892217139477526881306715143, 8.005691707083077555964883468116, 8.934927884322279446399086143565, 9.094741872857401152364938958964, 9.247545614676605058209101754583, 9.568211410120342513242644577001, 10.15761349996729779102090249759, 10.22062006650143921146447576915
