| L(s) = 1 | − 2.29·2-s + 3.25·4-s + 1.04·5-s + 1.05·7-s − 2.88·8-s − 2.40·10-s − 2.36·11-s + 2.07·13-s − 2.42·14-s + 0.107·16-s − 3.33·17-s + 1.50·19-s + 3.41·20-s + 5.42·22-s − 6.16·23-s − 3.90·25-s − 4.75·26-s + 3.44·28-s − 8.17·29-s − 0.508·31-s + 5.53·32-s + 7.65·34-s + 1.10·35-s + 2.24·37-s − 3.44·38-s − 3.02·40-s + 0.336·41-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 1.62·4-s + 0.468·5-s + 0.399·7-s − 1.02·8-s − 0.760·10-s − 0.713·11-s + 0.575·13-s − 0.647·14-s + 0.0268·16-s − 0.809·17-s + 0.345·19-s + 0.764·20-s + 1.15·22-s − 1.28·23-s − 0.780·25-s − 0.933·26-s + 0.650·28-s − 1.51·29-s − 0.0912·31-s + 0.978·32-s + 1.31·34-s + 0.187·35-s + 0.369·37-s − 0.559·38-s − 0.479·40-s + 0.0525·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 2 | \( 1 + 2.29T + 2T^{2} \) |
| 5 | \( 1 - 1.04T + 5T^{2} \) |
| 7 | \( 1 - 1.05T + 7T^{2} \) |
| 11 | \( 1 + 2.36T + 11T^{2} \) |
| 13 | \( 1 - 2.07T + 13T^{2} \) |
| 17 | \( 1 + 3.33T + 17T^{2} \) |
| 19 | \( 1 - 1.50T + 19T^{2} \) |
| 23 | \( 1 + 6.16T + 23T^{2} \) |
| 29 | \( 1 + 8.17T + 29T^{2} \) |
| 31 | \( 1 + 0.508T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 - 0.336T + 41T^{2} \) |
| 43 | \( 1 + 9.87T + 43T^{2} \) |
| 47 | \( 1 + 6.19T + 47T^{2} \) |
| 53 | \( 1 - 9.64T + 53T^{2} \) |
| 59 | \( 1 - 7.95T + 59T^{2} \) |
| 61 | \( 1 - 7.27T + 61T^{2} \) |
| 67 | \( 1 + 2.92T + 67T^{2} \) |
| 71 | \( 1 + 1.25T + 71T^{2} \) |
| 73 | \( 1 + 0.680T + 73T^{2} \) |
| 79 | \( 1 + 0.430T + 79T^{2} \) |
| 83 | \( 1 - 7.34T + 83T^{2} \) |
| 89 | \( 1 - 4.64T + 89T^{2} \) |
| 97 | \( 1 + 4.85T + 97T^{2} \) |
| 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
−9.278600395613119018567624067072, −8.353343087697839413525062014947, −7.935233849892802703315351641235, −7.02715296366830189671319735240, −6.14056969377399145355359000595, −5.20719979526978385122358136255, −3.87189371521771237830656563509, −2.35315214865105229736918718929, −1.60719520227465960785473144366, 0,
1.60719520227465960785473144366, 2.35315214865105229736918718929, 3.87189371521771237830656563509, 5.20719979526978385122358136255, 6.14056969377399145355359000595, 7.02715296366830189671319735240, 7.935233849892802703315351641235, 8.353343087697839413525062014947, 9.278600395613119018567624067072
