| L(s) = 1 | + 2·5-s + 2·7-s + 2·11-s − 2·13-s − 8·19-s − 6·23-s − 7·25-s − 10·29-s + 10·31-s + 4·35-s − 8·37-s − 14·41-s − 10·43-s + 2·47-s − 5·49-s − 8·53-s + 4·55-s − 14·59-s + 6·61-s − 4·65-s − 10·67-s − 4·71-s + 4·77-s + 22·79-s + 6·83-s − 16·89-s − 4·91-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 0.603·11-s − 0.554·13-s − 1.83·19-s − 1.25·23-s − 7/5·25-s − 1.85·29-s + 1.79·31-s + 0.676·35-s − 1.31·37-s − 2.18·41-s − 1.52·43-s + 0.291·47-s − 5/7·49-s − 1.09·53-s + 0.539·55-s − 1.82·59-s + 0.768·61-s − 0.496·65-s − 1.22·67-s − 0.474·71-s + 0.455·77-s + 2.47·79-s + 0.658·83-s − 1.69·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 17 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 49 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 81 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 41 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 98 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 113 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 107 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 105 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 22 T + 273 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 169 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 16 T + 218 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 171 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.011610246749906490895334891958, −7.926413685382763973036456177005, −7.31144338011062910065244253599, −6.86953852687247563894046562594, −6.42091505781869915447720083135, −6.35125090859543457081741332730, −5.91846055960653452877419964870, −5.55751681320044101093018990181, −4.92482838914868723279164283972, −4.90198581814807456780535441940, −4.36823002282681997660380240995, −3.94877760213740759056110670349, −3.49736183058929332828357681177, −3.19084127629040496665354144147, −2.21800268909282324256812118730, −2.17698492914301764935021916367, −1.61737496797709842929059814747, −1.50526010238665151867920340255, 0, 0,
1.50526010238665151867920340255, 1.61737496797709842929059814747, 2.17698492914301764935021916367, 2.21800268909282324256812118730, 3.19084127629040496665354144147, 3.49736183058929332828357681177, 3.94877760213740759056110670349, 4.36823002282681997660380240995, 4.90198581814807456780535441940, 4.92482838914868723279164283972, 5.55751681320044101093018990181, 5.91846055960653452877419964870, 6.35125090859543457081741332730, 6.42091505781869915447720083135, 6.86953852687247563894046562594, 7.31144338011062910065244253599, 7.926413685382763973036456177005, 8.011610246749906490895334891958
