| L(s) = 1 | + 2·2-s + 3·4-s + 2·5-s + 4·7-s + 4·8-s + 4·10-s − 2·11-s + 8·14-s + 5·16-s − 4·17-s + 6·19-s + 6·20-s − 4·22-s − 2·23-s + 12·28-s − 8·29-s + 8·31-s + 6·32-s − 8·34-s + 8·35-s − 2·37-s + 12·38-s + 8·40-s − 12·41-s + 6·43-s − 6·44-s − 4·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s + 1.51·7-s + 1.41·8-s + 1.26·10-s − 0.603·11-s + 2.13·14-s + 5/4·16-s − 0.970·17-s + 1.37·19-s + 1.34·20-s − 0.852·22-s − 0.417·23-s + 2.26·28-s − 1.48·29-s + 1.43·31-s + 1.06·32-s − 1.37·34-s + 1.35·35-s − 0.328·37-s + 1.94·38-s + 1.26·40-s − 1.87·41-s + 0.914·43-s − 0.904·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171396 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171396 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.396349627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.396349627\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 16 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 100 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 124 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 120 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 122 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 22 T + 280 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 182 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−11.43855571395339160704233000076, −11.24706591008395262448385903433, −10.68848734132717030147684526857, −10.26369283370254700325513181927, −9.691996385682068197400810950477, −9.373540314571360554561536372032, −8.457686863514327869607352044619, −8.213151869880932335026902257164, −7.58760820887045455896764147784, −7.28367338579452597907608231946, −6.34392027162519977867855247693, −6.33913434079100168180118776711, −5.39964865193001709794215659931, −5.13666143989143892882778907973, −4.85778267893092430970156153453, −4.15891181000549064408900710699, −3.43996159070286107653714567357, −2.75544910445994518040501664079, −1.96010573645468974781985746617, −1.57239268482570641614203851635,
1.57239268482570641614203851635, 1.96010573645468974781985746617, 2.75544910445994518040501664079, 3.43996159070286107653714567357, 4.15891181000549064408900710699, 4.85778267893092430970156153453, 5.13666143989143892882778907973, 5.39964865193001709794215659931, 6.33913434079100168180118776711, 6.34392027162519977867855247693, 7.28367338579452597907608231946, 7.58760820887045455896764147784, 8.213151869880932335026902257164, 8.457686863514327869607352044619, 9.373540314571360554561536372032, 9.691996385682068197400810950477, 10.26369283370254700325513181927, 10.68848734132717030147684526857, 11.24706591008395262448385903433, 11.43855571395339160704233000076
