| L(s) = 1 | − 0.317·3-s − 2.89·9-s + 3.78·11-s + 1.89·17-s − 5.97·19-s + 1.87·27-s − 1.20·33-s + 6.79·41-s − 8.48·43-s − 7·49-s − 0.603·51-s + 1.89·57-s + 14.1·59-s − 16.3·67-s − 15.6·73-s + 8.10·81-s + 17.0·83-s − 4.10·89-s − 10·97-s − 10.9·99-s + 15.0·107-s − 18.7·113-s + ⋯ |
| L(s) = 1 | − 0.183·3-s − 0.966·9-s + 1.14·11-s + 0.460·17-s − 1.37·19-s + 0.360·27-s − 0.209·33-s + 1.06·41-s − 1.29·43-s − 49-s − 0.0845·51-s + 0.251·57-s + 1.84·59-s − 1.99·67-s − 1.83·73-s + 0.900·81-s + 1.86·83-s − 0.434·89-s − 1.01·97-s − 1.10·99-s + 1.45·107-s − 1.76·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 0.317T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 3.78T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 1.89T + 17T^{2} \) |
| 19 | \( 1 + 5.97T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 6.79T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 16.3T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 + 4.10T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.72921369308427737460253147855, −6.80346248493212864104025836575, −6.27118766582814084051035294539, −5.66727363198885327708673664565, −4.77430953879237443375719650367, −4.01594161833870883294481919283, −3.23557334039959480290541199084, −2.30907265445140981471620614121, −1.28290468989391920260989659965, 0,
1.28290468989391920260989659965, 2.30907265445140981471620614121, 3.23557334039959480290541199084, 4.01594161833870883294481919283, 4.77430953879237443375719650367, 5.66727363198885327708673664565, 6.27118766582814084051035294539, 6.80346248493212864104025836575, 7.72921369308427737460253147855
