L(s) = 1 | − 3-s − 5-s − 2·9-s − 2·11-s + 15-s + 4·17-s + 2·19-s − 23-s + 25-s + 5·27-s + 9·29-s − 4·31-s + 2·33-s + 4·37-s + 41-s − 9·43-s + 2·45-s − 4·51-s − 10·53-s + 2·55-s − 2·57-s + 10·59-s + 9·61-s − 5·67-s + 69-s − 14·71-s + 12·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.603·11-s + 0.258·15-s + 0.970·17-s + 0.458·19-s − 0.208·23-s + 1/5·25-s + 0.962·27-s + 1.67·29-s − 0.718·31-s + 0.348·33-s + 0.657·37-s + 0.156·41-s − 1.37·43-s + 0.298·45-s − 0.560·51-s − 1.37·53-s + 0.269·55-s − 0.264·57-s + 1.30·59-s + 1.15·61-s − 0.610·67-s + 0.120·69-s − 1.66·71-s + 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.202452837462303423700669124048, −7.36767652168016478409515935302, −6.60848299643773930272929956285, −5.74860622941869667785795103711, −5.22074903874955150365006029497, −4.40545043539582889166562221247, −3.32113982539234444892411969160, −2.66814770361714405764404556770, −1.19775744210108337637987892371, 0,
1.19775744210108337637987892371, 2.66814770361714405764404556770, 3.32113982539234444892411969160, 4.40545043539582889166562221247, 5.22074903874955150365006029497, 5.74860622941869667785795103711, 6.60848299643773930272929956285, 7.36767652168016478409515935302, 8.202452837462303423700669124048