| L(s) = 1 | − 0.962·2-s + 2.64·3-s − 1.07·4-s + 5-s − 2.55·6-s − 3.09·7-s + 2.95·8-s + 4.01·9-s − 0.962·10-s − 6.13·11-s − 2.84·12-s + 1.16·13-s + 2.97·14-s + 2.64·15-s − 0.703·16-s − 3.86·18-s − 5.42·19-s − 1.07·20-s − 8.19·21-s + 5.90·22-s + 3.11·23-s + 7.83·24-s + 25-s − 1.11·26-s + 2.69·27-s + 3.32·28-s − 4.99·29-s + ⋯ |
| L(s) = 1 | − 0.680·2-s + 1.52·3-s − 0.536·4-s + 0.447·5-s − 1.04·6-s − 1.16·7-s + 1.04·8-s + 1.33·9-s − 0.304·10-s − 1.84·11-s − 0.820·12-s + 0.321·13-s + 0.796·14-s + 0.683·15-s − 0.175·16-s − 0.911·18-s − 1.24·19-s − 0.239·20-s − 1.78·21-s + 1.25·22-s + 0.650·23-s + 1.59·24-s + 0.200·25-s − 0.219·26-s + 0.518·27-s + 0.627·28-s − 0.927·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.962T + 2T^{2} \) |
| 3 | \( 1 - 2.64T + 3T^{2} \) |
| 7 | \( 1 + 3.09T + 7T^{2} \) |
| 11 | \( 1 + 6.13T + 11T^{2} \) |
| 13 | \( 1 - 1.16T + 13T^{2} \) |
| 19 | \( 1 + 5.42T + 19T^{2} \) |
| 23 | \( 1 - 3.11T + 23T^{2} \) |
| 29 | \( 1 + 4.99T + 29T^{2} \) |
| 31 | \( 1 + 3.72T + 31T^{2} \) |
| 37 | \( 1 - 0.396T + 37T^{2} \) |
| 41 | \( 1 + 1.70T + 41T^{2} \) |
| 43 | \( 1 + 0.0268T + 43T^{2} \) |
| 47 | \( 1 + 5.43T + 47T^{2} \) |
| 53 | \( 1 - 0.345T + 53T^{2} \) |
| 59 | \( 1 - 4.06T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 5.62T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 1.65T + 73T^{2} \) |
| 79 | \( 1 - 5.27T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 3.22T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
| 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
−9.059575685306240141389334722321, −8.507874122049819153609236198708, −7.80068654151212913473337897024, −7.08486388488023666421981477922, −5.86787345857544339204433122961, −4.79005348406458465524036582058, −3.66228877072266606422121041884, −2.85122272085309536310483905797, −1.90703864321361850110089219134, 0,
1.90703864321361850110089219134, 2.85122272085309536310483905797, 3.66228877072266606422121041884, 4.79005348406458465524036582058, 5.86787345857544339204433122961, 7.08486388488023666421981477922, 7.80068654151212913473337897024, 8.507874122049819153609236198708, 9.059575685306240141389334722321
