| L(s) = 1 | + 2·2-s − 6·3-s + 4-s + 24·5-s − 12·6-s + 12·8-s + 27·9-s + 48·10-s − 44·11-s − 6·12-s − 26·13-s − 144·15-s − 11·16-s + 164·17-s + 54·18-s + 48·19-s + 24·20-s − 88·22-s + 8·23-s − 72·24-s + 238·25-s − 52·26-s − 108·27-s + 404·29-s − 288·30-s + 40·31-s − 170·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/8·4-s + 2.14·5-s − 0.816·6-s + 0.530·8-s + 9-s + 1.51·10-s − 1.20·11-s − 0.144·12-s − 0.554·13-s − 2.47·15-s − 0.171·16-s + 2.33·17-s + 0.707·18-s + 0.579·19-s + 0.268·20-s − 0.852·22-s + 0.0725·23-s − 0.612·24-s + 1.90·25-s − 0.392·26-s − 0.769·27-s + 2.58·29-s − 1.75·30-s + 0.231·31-s − 0.939·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.091272112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.091272112\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 24 T + 338 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 90 p T^{2} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 p T + 1130 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 164 T + 16326 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 48 T + 14238 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T - 7906 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 404 T + 81518 T^{2} - 404 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 40 T + 50518 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 100 T + 92830 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 200 T + 24138 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 616 T + 216022 T^{2} + 616 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 324 T + 219554 T^{2} + 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 164 T + 102878 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 140 T + 393258 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 628 T + 293614 T^{2} - 628 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 472 T + 252622 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 428 T + 662834 T^{2} - 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 900 T + 899670 T^{2} + 900 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 432 T + 924318 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1388 T + 1567866 T^{2} + 1388 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 960 T + 1134938 T^{2} - 960 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 532 T + 1218502 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−16.35389087788887356414339717339, −15.59444456931606781246486698826, −14.39854807466564035287813392804, −14.31455700784265084109245368927, −13.56200222549742787414274124646, −13.16394236659639746642859752325, −12.61014534318416531167169799484, −11.98693603499908101350498142670, −11.25621080408741270288584986803, −10.22365105567801067018029684852, −10.00945851307149139236838779151, −9.829271318127610106617054814351, −8.365770672875975183148628021105, −7.43059213178317394275594944919, −6.51744942665318506834786953848, −5.85648758883860065584668602272, −5.05952133174969234512209771076, −4.99204993514547714957112126026, −2.95880254359165764848938604180, −1.51620518894178504511611455851,
1.51620518894178504511611455851, 2.95880254359165764848938604180, 4.99204993514547714957112126026, 5.05952133174969234512209771076, 5.85648758883860065584668602272, 6.51744942665318506834786953848, 7.43059213178317394275594944919, 8.365770672875975183148628021105, 9.829271318127610106617054814351, 10.00945851307149139236838779151, 10.22365105567801067018029684852, 11.25621080408741270288584986803, 11.98693603499908101350498142670, 12.61014534318416531167169799484, 13.16394236659639746642859752325, 13.56200222549742787414274124646, 14.31455700784265084109245368927, 14.39854807466564035287813392804, 15.59444456931606781246486698826, 16.35389087788887356414339717339
