L(s) = 1 | − 2-s + 2.13·3-s + 4-s + 0.446·5-s − 2.13·6-s − 1.57·7-s − 8-s + 1.54·9-s − 0.446·10-s − 4.27·11-s + 2.13·12-s − 6.29·13-s + 1.57·14-s + 0.952·15-s + 16-s + 0.352·17-s − 1.54·18-s + 6.75·19-s + 0.446·20-s − 3.36·21-s + 4.27·22-s − 1.57·23-s − 2.13·24-s − 4.80·25-s + 6.29·26-s − 3.10·27-s − 1.57·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.23·3-s + 0.5·4-s + 0.199·5-s − 0.869·6-s − 0.596·7-s − 0.353·8-s + 0.513·9-s − 0.141·10-s − 1.28·11-s + 0.615·12-s − 1.74·13-s + 0.421·14-s + 0.245·15-s + 0.250·16-s + 0.0854·17-s − 0.363·18-s + 1.54·19-s + 0.0999·20-s − 0.733·21-s + 0.911·22-s − 0.327·23-s − 0.434·24-s − 0.960·25-s + 1.23·26-s − 0.598·27-s − 0.298·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 - 2.13T + 3T^{2} \) |
| 5 | \( 1 - 0.446T + 5T^{2} \) |
| 7 | \( 1 + 1.57T + 7T^{2} \) |
| 11 | \( 1 + 4.27T + 11T^{2} \) |
| 13 | \( 1 + 6.29T + 13T^{2} \) |
| 17 | \( 1 - 0.352T + 17T^{2} \) |
| 19 | \( 1 - 6.75T + 19T^{2} \) |
| 23 | \( 1 + 1.57T + 23T^{2} \) |
| 29 | \( 1 + 1.26T + 29T^{2} \) |
| 31 | \( 1 + 5.51T + 31T^{2} \) |
| 37 | \( 1 - 4.73T + 37T^{2} \) |
| 41 | \( 1 - 3.24T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 - 2.12T + 47T^{2} \) |
| 53 | \( 1 - 2.42T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 2.74T + 61T^{2} \) |
| 67 | \( 1 - 4.64T + 67T^{2} \) |
| 71 | \( 1 - 9.09T + 71T^{2} \) |
| 73 | \( 1 + 4.43T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 + 4.87T + 83T^{2} \) |
| 89 | \( 1 + 1.72T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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
−9.359009926785572506155897593195, −8.144984929349601227413505040438, −7.69146549742860352282434759120, −7.12648816128736580635037507609, −5.81452768648207405082406861342, −4.97358341237569574246844803390, −3.46832275144324940258307867382, −2.75677683343591506794633609296, −2.00046676742539989282627255128, 0,
2.00046676742539989282627255128, 2.75677683343591506794633609296, 3.46832275144324940258307867382, 4.97358341237569574246844803390, 5.81452768648207405082406861342, 7.12648816128736580635037507609, 7.69146549742860352282434759120, 8.144984929349601227413505040438, 9.359009926785572506155897593195