| L(s) = 1 | − 2·5-s − 7-s + 2·11-s − 4·13-s + 4·17-s − 19-s − 25-s + 2·29-s + 10·31-s + 2·35-s + 4·37-s − 6·41-s − 12·43-s − 8·47-s + 49-s + 6·53-s − 4·55-s − 2·61-s + 8·65-s + 4·67-s + 14·73-s − 2·77-s − 2·79-s + 6·83-s − 8·85-s − 14·89-s + 4·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 0.603·11-s − 1.10·13-s + 0.970·17-s − 0.229·19-s − 1/5·25-s + 0.371·29-s + 1.79·31-s + 0.338·35-s + 0.657·37-s − 0.937·41-s − 1.82·43-s − 1.16·47-s + 1/7·49-s + 0.824·53-s − 0.539·55-s − 0.256·61-s + 0.992·65-s + 0.488·67-s + 1.63·73-s − 0.227·77-s − 0.225·79-s + 0.658·83-s − 0.867·85-s − 1.48·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.34837345937667523328253756652, −6.70998628944307387114146719421, −6.14757994741989338568062344505, −5.09988363916136829570686585515, −4.61574387960280580811398526547, −3.72536075438044696576364080535, −3.18374261308210579055293179844, −2.26042581427098772187271075375, −1.07786205982421442325699058420, 0,
1.07786205982421442325699058420, 2.26042581427098772187271075375, 3.18374261308210579055293179844, 3.72536075438044696576364080535, 4.61574387960280580811398526547, 5.09988363916136829570686585515, 6.14757994741989338568062344505, 6.70998628944307387114146719421, 7.34837345937667523328253756652
