| L(s) = 1 | − 4·7-s − 9·9-s − 50·25-s − 92·31-s − 86·49-s + 36·63-s + 92·73-s − 284·79-s + 81·81-s + 4·97-s − 388·103-s − 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 146·169-s + 173-s + 200·175-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 4/7·7-s − 9-s − 2·25-s − 2.96·31-s − 1.75·49-s + 4/7·63-s + 1.26·73-s − 3.59·79-s + 81-s + 4/97·97-s − 3.76·103-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.863·169-s + 0.00578·173-s + 8/7·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.05052142868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05052142868\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 146 T^{2} + p^{4} T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 46 T^{2} + p^{4} T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2062 T^{2} + p^{4} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3214 T^{2} + p^{4} T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 1966 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 5906 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 142 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
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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
−10.69901461569784638289812565629, −9.622497719935074179156296211670, −9.458238681259721737107403775751, −9.357660586209115886535279899535, −8.465810867008295580775212074127, −8.310134953272945720365024022204, −7.83813511511653399244683922728, −7.11529539740192100346029068721, −7.08546330163177382204926681885, −6.25025281573276720863993967569, −5.88484795060959179149772808357, −5.53542024653655314612639049616, −5.12016377703700012228855489155, −4.30381774664298985482186582371, −3.79827736007688958404857099721, −3.38044693374623922639561692672, −2.79309353029475354324169064819, −2.06779576055598114317599754321, −1.49925414073182010686894721462, −0.07421213470976312642838662138,
0.07421213470976312642838662138, 1.49925414073182010686894721462, 2.06779576055598114317599754321, 2.79309353029475354324169064819, 3.38044693374623922639561692672, 3.79827736007688958404857099721, 4.30381774664298985482186582371, 5.12016377703700012228855489155, 5.53542024653655314612639049616, 5.88484795060959179149772808357, 6.25025281573276720863993967569, 7.08546330163177382204926681885, 7.11529539740192100346029068721, 7.83813511511653399244683922728, 8.310134953272945720365024022204, 8.465810867008295580775212074127, 9.357660586209115886535279899535, 9.458238681259721737107403775751, 9.622497719935074179156296211670, 10.69901461569784638289812565629
