L(s) = 1 | + 1.39·2-s − 1.21·3-s − 0.0545·4-s − 0.0316·5-s − 1.69·6-s − 2.20·7-s − 2.86·8-s − 1.52·9-s − 0.0441·10-s − 2.27·11-s + 0.0662·12-s − 13-s − 3.07·14-s + 0.0384·15-s − 3.88·16-s − 0.756·17-s − 2.12·18-s + 6.64·19-s + 0.00172·20-s + 2.67·21-s − 3.16·22-s − 2.39·23-s + 3.48·24-s − 4.99·25-s − 1.39·26-s + 5.49·27-s + 0.120·28-s + ⋯ |
L(s) = 1 | + 0.986·2-s − 0.701·3-s − 0.0272·4-s − 0.0141·5-s − 0.691·6-s − 0.832·7-s − 1.01·8-s − 0.508·9-s − 0.0139·10-s − 0.684·11-s + 0.0191·12-s − 0.277·13-s − 0.821·14-s + 0.00992·15-s − 0.971·16-s − 0.183·17-s − 0.501·18-s + 1.52·19-s + 0.000386·20-s + 0.584·21-s − 0.675·22-s − 0.499·23-s + 0.710·24-s − 0.999·25-s − 0.273·26-s + 1.05·27-s + 0.0227·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 3 | \( 1 + 1.21T + 3T^{2} \) |
| 5 | \( 1 + 0.0316T + 5T^{2} \) |
| 7 | \( 1 + 2.20T + 7T^{2} \) |
| 11 | \( 1 + 2.27T + 11T^{2} \) |
| 17 | \( 1 + 0.756T + 17T^{2} \) |
| 19 | \( 1 - 6.64T + 19T^{2} \) |
| 23 | \( 1 + 2.39T + 23T^{2} \) |
| 29 | \( 1 + 2.98T + 29T^{2} \) |
| 37 | \( 1 + 7.31T + 37T^{2} \) |
| 41 | \( 1 - 1.75T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 4.81T + 53T^{2} \) |
| 59 | \( 1 - 9.24T + 59T^{2} \) |
| 61 | \( 1 + 7.71T + 61T^{2} \) |
| 67 | \( 1 + 3.49T + 67T^{2} \) |
| 71 | \( 1 - 6.25T + 71T^{2} \) |
| 73 | \( 1 + 1.71T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 4.36T + 83T^{2} \) |
| 89 | \( 1 - 0.704T + 89T^{2} \) |
| 97 | \( 1 - 1.02T + 97T^{2} \) |
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
−11.09249683669521387375520124590, −9.925662020600416174992675188164, −9.175264133801372099388551838223, −7.88618030415170438584494971230, −6.64230820614375607002598570972, −5.67446649243988261246112218992, −5.17982604372843120280706097640, −3.81528894761928973337490945168, −2.79025995339998899099059676332, 0,
2.79025995339998899099059676332, 3.81528894761928973337490945168, 5.17982604372843120280706097640, 5.67446649243988261246112218992, 6.64230820614375607002598570972, 7.88618030415170438584494971230, 9.175264133801372099388551838223, 9.925662020600416174992675188164, 11.09249683669521387375520124590