| L(s) = 1 | − 2·3-s − 2·4-s − 5-s − 4·7-s + 9-s + 3·11-s + 4·12-s + 2·13-s + 2·15-s + 4·16-s + 6·17-s + 2·20-s + 8·21-s + 25-s + 4·27-s + 8·28-s − 3·29-s − 7·31-s − 6·33-s + 4·35-s − 2·36-s + 8·37-s − 4·39-s − 6·41-s − 4·43-s − 6·44-s − 45-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.904·11-s + 1.15·12-s + 0.554·13-s + 0.516·15-s + 16-s + 1.45·17-s + 0.447·20-s + 1.74·21-s + 1/5·25-s + 0.769·27-s + 1.51·28-s − 0.557·29-s − 1.25·31-s − 1.04·33-s + 0.676·35-s − 1/3·36-s + 1.31·37-s − 0.640·39-s − 0.937·41-s − 0.609·43-s − 0.904·44-s − 0.149·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.127628990679212829761037204252, −8.125110666225905114287996427728, −7.15688226851931363649415056933, −6.21260132573943813797134499627, −5.78864269288415358256356915363, −4.85687055939197751337392409509, −3.76032779270688982178507540089, −3.30897413440669327666321670557, −1.06417714624828988770031977779, 0,
1.06417714624828988770031977779, 3.30897413440669327666321670557, 3.76032779270688982178507540089, 4.85687055939197751337392409509, 5.78864269288415358256356915363, 6.21260132573943813797134499627, 7.15688226851931363649415056933, 8.125110666225905114287996427728, 9.127628990679212829761037204252
