| L(s) = 1 | − 8·2-s + 18·3-s + 48·4-s + 17·5-s − 144·6-s − 256·8-s + 243·9-s − 136·10-s + 145·11-s + 864·12-s − 715·13-s + 306·15-s + 1.28e3·16-s + 1.37e3·17-s − 1.94e3·18-s + 1.08e3·19-s + 816·20-s − 1.16e3·22-s − 4.50e3·23-s − 4.60e3·24-s + 153·25-s + 5.72e3·26-s + 2.91e3·27-s + 7.86e3·29-s − 2.44e3·30-s + 8.81e3·31-s − 6.14e3·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.304·5-s − 1.63·6-s − 1.41·8-s + 9-s − 0.430·10-s + 0.361·11-s + 1.73·12-s − 1.17·13-s + 0.351·15-s + 5/4·16-s + 1.15·17-s − 1.41·18-s + 0.686·19-s + 0.456·20-s − 0.510·22-s − 1.77·23-s − 1.63·24-s + 0.0489·25-s + 1.65·26-s + 0.769·27-s + 1.73·29-s − 0.496·30-s + 1.64·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.266386709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.266386709\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 17 T + 136 T^{2} - 17 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 145 T + 24232 T^{2} - 145 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 55 p T + 864206 T^{2} + 55 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1372 T + 835810 T^{2} - 1372 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1081 T + 734562 T^{2} - 1081 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 196 p T + 17854222 T^{2} + 196 p^{6} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 7865 T + 34952518 T^{2} - 7865 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8816 T + 75476261 T^{2} - 8816 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14573 T + 151193010 T^{2} + 14573 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7350 T + 218270722 T^{2} - 7350 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5921 T + 278228220 T^{2} + 5921 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 44808 T + 952611850 T^{2} - 44808 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9417 T + 846033802 T^{2} - 9417 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5077 T + 557047054 T^{2} + 5077 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 42368 T + 2085594038 T^{2} - 42368 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 30501 T + 2311177888 T^{2} - 30501 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 91744 T + 5413284586 T^{2} - 91744 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 85665 T + 4403117836 T^{2} + 85665 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 94646 T + 5465354807 T^{2} - 94646 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 33841 T + 8158439620 T^{2} - 33841 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 27558 T + 11170684834 T^{2} - 27558 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 46671 T + 13111898848 T^{2} + 46671 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.87463739133645118651468807411, −10.28932975506884922765670673827, −9.995071171127754295033392699745, −9.884156297753468753058357369687, −9.305633130434703988421175987531, −8.741578811171074533806265761212, −8.343529862145523721719299175462, −8.013581811033491124498321878167, −7.37247997590109033577052668590, −7.16879695577871465880760137368, −6.46677467970899495239820679570, −5.91099618663856588990856296644, −5.20063002141894807354004453609, −4.46157719735588860176666285992, −3.64468965828009827715068435231, −3.10359801505442478026714187772, −2.22893491864973219912472689165, −2.18705096190296499568794036170, −1.01989775413966587737934946304, −0.68895885433028725289820913320,
0.68895885433028725289820913320, 1.01989775413966587737934946304, 2.18705096190296499568794036170, 2.22893491864973219912472689165, 3.10359801505442478026714187772, 3.64468965828009827715068435231, 4.46157719735588860176666285992, 5.20063002141894807354004453609, 5.91099618663856588990856296644, 6.46677467970899495239820679570, 7.16879695577871465880760137368, 7.37247997590109033577052668590, 8.013581811033491124498321878167, 8.343529862145523721719299175462, 8.741578811171074533806265761212, 9.305633130434703988421175987531, 9.884156297753468753058357369687, 9.995071171127754295033392699745, 10.28932975506884922765670673827, 10.87463739133645118651468807411
