| L(s) = 1 | − 2·2-s − 10·3-s − 2·4-s + 20·6-s + 21·9-s + 66·11-s + 20·12-s − 10·13-s − 20·16-s − 70·17-s − 42·18-s + 140·19-s − 132·22-s + 16·23-s + 20·26-s + 310·27-s − 258·29-s − 20·31-s + 200·32-s − 660·33-s + 140·34-s − 42·36-s − 328·37-s − 280·38-s + 100·39-s + 300·41-s + 116·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.92·3-s − 1/4·4-s + 1.36·6-s + 7/9·9-s + 1.80·11-s + 0.481·12-s − 0.213·13-s − 0.312·16-s − 0.998·17-s − 0.549·18-s + 1.69·19-s − 1.27·22-s + 0.145·23-s + 0.150·26-s + 2.20·27-s − 1.65·29-s − 0.115·31-s + 1.10·32-s − 3.48·33-s + 0.706·34-s − 0.194·36-s − 1.45·37-s − 1.19·38-s + 0.410·39-s + 1.14·41-s + 0.411·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 p T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + 5 T + p^{3} T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 6 p T + 325 p T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 10 T + 19 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 70 T + 6651 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 140 T + 14218 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 16 T + 15774 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 258 T + 40075 T^{2} + 258 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 20 T + 20082 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 328 T + 106906 T^{2} + 328 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 300 T + 50342 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 116 T + 160794 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 30 T + 190271 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 540 T + 212254 T^{2} + 540 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 380 T + 429258 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1080 T + 705962 T^{2} - 1080 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 468 T + 554906 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1056 T + 949550 T^{2} + 1056 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 860 T + 522934 T^{2} - 860 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 p T - 339825 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 40 T + 703974 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 240 T - 164062 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1630 T + 2133171 T^{2} + 1630 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.337339938333790283031126639738, −8.891927907879713775258652473462, −8.339348897343329050629923860150, −8.151896113061880260420965275303, −7.21949170855399631477235080772, −6.93849799901040264460646550767, −6.76132982819556586666561363676, −6.18996418187791878668081681433, −5.68837466709524637653645616353, −5.53008654493037169805523538719, −5.05343563634502871415873654645, −4.45800634127096850321536831286, −4.09293973645410841373521660606, −3.45663127516272014003248717196, −2.85587822509341887217602807610, −2.08833329092784406373988128802, −1.21300579946368877754113670236, −0.926968934824066914121227996689, 0, 0,
0.926968934824066914121227996689, 1.21300579946368877754113670236, 2.08833329092784406373988128802, 2.85587822509341887217602807610, 3.45663127516272014003248717196, 4.09293973645410841373521660606, 4.45800634127096850321536831286, 5.05343563634502871415873654645, 5.53008654493037169805523538719, 5.68837466709524637653645616353, 6.18996418187791878668081681433, 6.76132982819556586666561363676, 6.93849799901040264460646550767, 7.21949170855399631477235080772, 8.151896113061880260420965275303, 8.339348897343329050629923860150, 8.891927907879713775258652473462, 9.337339938333790283031126639738
