L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 4·11-s − 2·13-s − 15-s + 2·17-s + 4·19-s + 21-s + 23-s + 25-s − 27-s − 2·29-s − 8·31-s − 4·33-s − 35-s + 6·37-s + 2·39-s − 6·41-s + 4·43-s + 45-s − 8·47-s + 49-s − 2·51-s − 10·53-s + 4·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s − 0.169·35-s + 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.280·51-s − 1.37·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.88922558603431, −14.51313702624966, −14.25459167011822, −13.39439686222563, −12.92012126217907, −12.59036503223648, −11.75822347872491, −11.55594951981439, −11.03799970631278, −10.07084781014313, −9.962461633048714, −9.288034532082561, −8.932581438756357, −8.051463988522835, −7.409804390718173, −6.868064418785684, −6.464876226322000, −5.652143194158785, −5.397214274788554, −4.626244781818530, −3.882809348736449, −3.358362411772085, −2.535951196285756, −1.631893230937162, −1.063497548545317, 0,
1.063497548545317, 1.631893230937162, 2.535951196285756, 3.358362411772085, 3.882809348736449, 4.626244781818530, 5.397214274788554, 5.652143194158785, 6.464876226322000, 6.868064418785684, 7.409804390718173, 8.051463988522835, 8.932581438756357, 9.288034532082561, 9.962461633048714, 10.07084781014313, 11.03799970631278, 11.55594951981439, 11.75822347872491, 12.59036503223648, 12.92012126217907, 13.39439686222563, 14.25459167011822, 14.51313702624966, 14.88922558603431