| L(s) = 1 | + 9-s − 8·11-s − 4·13-s + 8·19-s + 8·23-s − 6·25-s + 12·29-s − 12·41-s − 8·43-s − 14·49-s + 8·67-s + 20·73-s + 16·79-s + 81-s + 8·83-s − 8·99-s − 36·101-s − 32·103-s + 24·107-s − 4·117-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 32·143-s + 149-s + ⋯ |
| L(s) = 1 | + 1/3·9-s − 2.41·11-s − 1.10·13-s + 1.83·19-s + 1.66·23-s − 6/5·25-s + 2.22·29-s − 1.87·41-s − 1.21·43-s − 2·49-s + 0.977·67-s + 2.34·73-s + 1.80·79-s + 1/9·81-s + 0.878·83-s − 0.804·99-s − 3.58·101-s − 3.15·103-s + 2.32·107-s − 0.369·117-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.67·143-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 609408 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 609408 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 23 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.173282349834764667604075089133, −7.932741438944089455498075047781, −7.26958585488936859387169650645, −6.81091891714363813978370387900, −6.61894821311647461036273237236, −5.68693030173540232457460232898, −5.08637437240091893034392278396, −5.01357758335939619070346679818, −4.82931243251591807244664468719, −3.66511529305920391284937215509, −3.14826068230388029805668446658, −2.74128373145577575704343979657, −2.15578307237422591548694616714, −1.13850789947910914060996228408, 0,
1.13850789947910914060996228408, 2.15578307237422591548694616714, 2.74128373145577575704343979657, 3.14826068230388029805668446658, 3.66511529305920391284937215509, 4.82931243251591807244664468719, 5.01357758335939619070346679818, 5.08637437240091893034392278396, 5.68693030173540232457460232898, 6.61894821311647461036273237236, 6.81091891714363813978370387900, 7.26958585488936859387169650645, 7.932741438944089455498075047781, 8.173282349834764667604075089133
