| L(s) = 1 | − 20·25-s − 8·43-s + 14·49-s + 40·67-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 4·25-s − 1.21·43-s + 2·49-s + 4.88·67-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.632920383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.632920383\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.66903958824944054193687902986, −6.14367247025062897931291959614, −6.10469249020468045272436133087, −5.77288561825969586990546426220, −5.69439883689886087896797552360, −5.58588717345318452825639142773, −5.25079448165160186816799788248, −5.04911509464592722301570689035, −4.87235672784207207488981321064, −4.55251982046110318154268946655, −4.25358068751243185118736376574, −4.08142049499816751213882695477, −3.76178316227743244457026133585, −3.70567684302734585688699323347, −3.61412335200424185733540539726, −3.27543462056172750809678679863, −2.80929237746960930273109541457, −2.53693476061501171017310674369, −2.34041006185620916994675344043, −2.07452662080555410033685225157, −1.93413199320177566586340420702, −1.43829230817890212734702552232, −1.26238614712746558048647975995, −0.49480803358188047972852935507, −0.44053485531030557679731665682,
0.44053485531030557679731665682, 0.49480803358188047972852935507, 1.26238614712746558048647975995, 1.43829230817890212734702552232, 1.93413199320177566586340420702, 2.07452662080555410033685225157, 2.34041006185620916994675344043, 2.53693476061501171017310674369, 2.80929237746960930273109541457, 3.27543462056172750809678679863, 3.61412335200424185733540539726, 3.70567684302734585688699323347, 3.76178316227743244457026133585, 4.08142049499816751213882695477, 4.25358068751243185118736376574, 4.55251982046110318154268946655, 4.87235672784207207488981321064, 5.04911509464592722301570689035, 5.25079448165160186816799788248, 5.58588717345318452825639142773, 5.69439883689886087896797552360, 5.77288561825969586990546426220, 6.10469249020468045272436133087, 6.14367247025062897931291959614, 6.66903958824944054193687902986
