| L(s) = 1 | + 3-s − 5-s + 7-s − 2·9-s + 2·11-s + 13-s − 15-s + 17-s − 8·19-s + 21-s − 6·23-s − 4·25-s − 5·27-s + 6·29-s + 8·31-s + 2·33-s − 35-s − 37-s + 39-s − 4·41-s − 5·43-s + 2·45-s − 11·47-s − 6·49-s + 51-s − 6·53-s − 2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.603·11-s + 0.277·13-s − 0.258·15-s + 0.242·17-s − 1.83·19-s + 0.218·21-s − 1.25·23-s − 4/5·25-s − 0.962·27-s + 1.11·29-s + 1.43·31-s + 0.348·33-s − 0.169·35-s − 0.164·37-s + 0.160·39-s − 0.624·41-s − 0.762·43-s + 0.298·45-s − 1.60·47-s − 6/7·49-s + 0.140·51-s − 0.824·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 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
−8.271821286437898207293657776000, −7.904402555218795050388581979990, −6.55178838925157139920551369496, −6.27310404871084208164053894849, −5.08348067840021732049106929757, −4.21395480665120477666519490165, −3.57116491479007763423843388479, −2.55214837136789264020550060654, −1.63107791127337548543560799803, 0,
1.63107791127337548543560799803, 2.55214837136789264020550060654, 3.57116491479007763423843388479, 4.21395480665120477666519490165, 5.08348067840021732049106929757, 6.27310404871084208164053894849, 6.55178838925157139920551369496, 7.904402555218795050388581979990, 8.271821286437898207293657776000
