| L(s) = 1 | − 3-s − 5-s + 3·7-s + 9-s − 5·11-s + 2·13-s + 15-s − 17-s + 19-s − 3·21-s − 4·23-s − 4·25-s − 27-s + 6·29-s + 10·31-s + 5·33-s − 3·35-s − 2·39-s − 11·43-s − 45-s − 9·47-s + 2·49-s + 51-s − 10·53-s + 5·55-s − 57-s + 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s − 1.50·11-s + 0.554·13-s + 0.258·15-s − 0.242·17-s + 0.229·19-s − 0.654·21-s − 0.834·23-s − 4/5·25-s − 0.192·27-s + 1.11·29-s + 1.79·31-s + 0.870·33-s − 0.507·35-s − 0.320·39-s − 1.67·43-s − 0.149·45-s − 1.31·47-s + 2/7·49-s + 0.140·51-s − 1.37·53-s + 0.674·55-s − 0.132·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.137259653188121390855447482027, −7.65627920571478032514336271694, −6.60850791131037206491890289680, −5.90641506250154868855227026945, −4.90222652055451670072407703452, −4.68719487779910370709247985620, −3.52150549884707699550976993427, −2.43726074335480144289137857493, −1.36309445163470127656106423645, 0,
1.36309445163470127656106423645, 2.43726074335480144289137857493, 3.52150549884707699550976993427, 4.68719487779910370709247985620, 4.90222652055451670072407703452, 5.90641506250154868855227026945, 6.60850791131037206491890289680, 7.65627920571478032514336271694, 8.137259653188121390855447482027
