| L(s) = 1 | + 2·3-s − 2·5-s − 2·7-s + 3·9-s + 4·11-s − 2·13-s − 4·15-s + 6·17-s + 2·19-s − 4·21-s + 2·23-s + 3·25-s + 4·27-s + 6·31-s + 8·33-s + 4·35-s + 2·37-s − 4·39-s − 4·41-s + 4·43-s − 6·45-s + 18·47-s + 6·49-s + 12·51-s − 20·53-s − 8·55-s + 4·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.755·7-s + 9-s + 1.20·11-s − 0.554·13-s − 1.03·15-s + 1.45·17-s + 0.458·19-s − 0.872·21-s + 0.417·23-s + 3/5·25-s + 0.769·27-s + 1.07·31-s + 1.39·33-s + 0.676·35-s + 0.328·37-s − 0.640·39-s − 0.624·41-s + 0.609·43-s − 0.894·45-s + 2.62·47-s + 6/7·49-s + 1.68·51-s − 2.74·53-s − 1.07·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.743133620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.743133620\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18 T + 158 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.336349536537910038143027457836, −9.105545942976600031136106104062, −8.548614868626765517105458638044, −8.204122165867334562663232104127, −7.923661772558857739034943547920, −7.48590223665446113984640142768, −7.04383997978639446607404034692, −6.90326054952081759844647982283, −6.29185944526145201298811444519, −5.95270342377244980268667956690, −5.13277601610183396734163131203, −5.03933271199184502142709976957, −4.11812084680580603800069282974, −4.00067648987351164176442201733, −3.57023542982558475306107602630, −3.08277472450982197023974961367, −2.70466211496855052330750507842, −2.12064800997391153651051560447, −1.16438398777634297975299014551, −0.77836508983065244974024506531,
0.77836508983065244974024506531, 1.16438398777634297975299014551, 2.12064800997391153651051560447, 2.70466211496855052330750507842, 3.08277472450982197023974961367, 3.57023542982558475306107602630, 4.00067648987351164176442201733, 4.11812084680580603800069282974, 5.03933271199184502142709976957, 5.13277601610183396734163131203, 5.95270342377244980268667956690, 6.29185944526145201298811444519, 6.90326054952081759844647982283, 7.04383997978639446607404034692, 7.48590223665446113984640142768, 7.923661772558857739034943547920, 8.204122165867334562663232104127, 8.548614868626765517105458638044, 9.105545942976600031136106104062, 9.336349536537910038143027457836
