| L(s) = 1 | − 2·4-s + 3.19·7-s − 0.990·13-s + 4·16-s − 4.28·19-s − 6.38·28-s − 10.3·31-s + 12.1·37-s − 11.1·43-s + 3.19·49-s + 1.98·52-s + 4.21·61-s − 8·64-s − 13.2·67-s + 10.0·73-s + 8.56·76-s − 17.7·79-s − 3.16·91-s + 19.6·97-s − 14.1·103-s + 20.4·109-s + 12.7·112-s + ⋯ |
| L(s) = 1 | − 4-s + 1.20·7-s − 0.274·13-s + 16-s − 0.982·19-s − 1.20·28-s − 1.86·31-s + 1.99·37-s − 1.70·43-s + 0.455·49-s + 0.274·52-s + 0.540·61-s − 64-s − 1.62·67-s + 1.18·73-s + 0.982·76-s − 1.99·79-s − 0.331·91-s + 1.99·97-s − 1.39·103-s + 1.95·109-s + 1.20·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 - 3.19T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.990T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 4.28T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 12.1T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 4.21T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + 17.7T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 19.6T + 97T^{2} \) |
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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
−7.888914309317099574698382318498, −7.29274977588659062423782453959, −6.22120289860804258157620636411, −5.46668982018529211258191138522, −4.77958367796916064967879100577, −4.27567894726716329887693671627, −3.42618490871176329826144044406, −2.22025756161214952597911283040, −1.31273751195925041017508851642, 0,
1.31273751195925041017508851642, 2.22025756161214952597911283040, 3.42618490871176329826144044406, 4.27567894726716329887693671627, 4.77958367796916064967879100577, 5.46668982018529211258191138522, 6.22120289860804258157620636411, 7.29274977588659062423782453959, 7.888914309317099574698382318498
