L(s) = 1 | + 5-s + 7-s + 6·11-s − 2·13-s − 6·19-s − 23-s + 25-s + 4·29-s − 2·31-s + 35-s − 6·37-s − 6·41-s − 6·43-s + 2·47-s + 49-s + 2·53-s + 6·55-s + 4·59-s − 2·61-s − 2·65-s + 2·67-s + 10·71-s − 6·73-s + 6·77-s − 6·83-s − 10·89-s − 2·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.80·11-s − 0.554·13-s − 1.37·19-s − 0.208·23-s + 1/5·25-s + 0.742·29-s − 0.359·31-s + 0.169·35-s − 0.986·37-s − 0.937·41-s − 0.914·43-s + 0.291·47-s + 1/7·49-s + 0.274·53-s + 0.809·55-s + 0.520·59-s − 0.256·61-s − 0.248·65-s + 0.244·67-s + 1.18·71-s − 0.702·73-s + 0.683·77-s − 0.658·83-s − 1.05·89-s − 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.87632111703539, −13.48952203134021, −12.79018370766648, −12.31425402791057, −11.99403718215599, −11.43698518962249, −10.99362979985532, −10.35273231248352, −9.953026712423465, −9.475066313579985, −8.809752394511699, −8.572180144603959, −8.078067035712978, −7.060529169435856, −6.971189226004025, −6.345890857514572, −5.897004727628081, −5.173604952326838, −4.667538098811424, −4.096934966829665, −3.627944478669822, −2.887161141972489, −2.045913718263965, −1.721289881614764, −0.9789578358016160, 0,
0.9789578358016160, 1.721289881614764, 2.045913718263965, 2.887161141972489, 3.627944478669822, 4.096934966829665, 4.667538098811424, 5.173604952326838, 5.897004727628081, 6.345890857514572, 6.971189226004025, 7.060529169435856, 8.078067035712978, 8.572180144603959, 8.809752394511699, 9.475066313579985, 9.953026712423465, 10.35273231248352, 10.99362979985532, 11.43698518962249, 11.99403718215599, 12.31425402791057, 12.79018370766648, 13.48952203134021, 13.87632111703539