| L(s) = 1 | − 2.12·2-s − 1.84·3-s + 2.52·4-s + 3.91·6-s + 1.46·7-s − 1.12·8-s + 0.389·9-s + 5.01·11-s − 4.65·12-s + 5.11·13-s − 3.11·14-s − 2.66·16-s + 2.57·17-s − 0.829·18-s + 7.12·19-s − 2.69·21-s − 10.6·22-s + 1.20·23-s + 2.07·24-s − 10.8·26-s + 4.80·27-s + 3.69·28-s − 9.83·29-s + 6.53·31-s + 7.91·32-s − 9.23·33-s − 5.47·34-s + ⋯ |
| L(s) = 1 | − 1.50·2-s − 1.06·3-s + 1.26·4-s + 1.59·6-s + 0.552·7-s − 0.397·8-s + 0.129·9-s + 1.51·11-s − 1.34·12-s + 1.41·13-s − 0.831·14-s − 0.665·16-s + 0.624·17-s − 0.195·18-s + 1.63·19-s − 0.587·21-s − 2.27·22-s + 0.251·23-s + 0.423·24-s − 2.13·26-s + 0.924·27-s + 0.698·28-s − 1.82·29-s + 1.17·31-s + 1.39·32-s − 1.60·33-s − 0.939·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7103037788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7103037788\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 + 2.12T + 2T^{2} \) |
| 3 | \( 1 + 1.84T + 3T^{2} \) |
| 7 | \( 1 - 1.46T + 7T^{2} \) |
| 11 | \( 1 - 5.01T + 11T^{2} \) |
| 13 | \( 1 - 5.11T + 13T^{2} \) |
| 17 | \( 1 - 2.57T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 - 1.20T + 23T^{2} \) |
| 29 | \( 1 + 9.83T + 29T^{2} \) |
| 31 | \( 1 - 6.53T + 31T^{2} \) |
| 37 | \( 1 + 0.541T + 37T^{2} \) |
| 41 | \( 1 - 9.87T + 41T^{2} \) |
| 43 | \( 1 - 0.997T + 43T^{2} \) |
| 47 | \( 1 + 7.22T + 47T^{2} \) |
| 53 | \( 1 + 4.30T + 53T^{2} \) |
| 61 | \( 1 - 4.14T + 61T^{2} \) |
| 67 | \( 1 + 7.07T + 67T^{2} \) |
| 71 | \( 1 + 6.60T + 71T^{2} \) |
| 73 | \( 1 + 4.00T + 73T^{2} \) |
| 79 | \( 1 + 2.03T + 79T^{2} \) |
| 83 | \( 1 + 6.02T + 83T^{2} \) |
| 89 | \( 1 - 0.199T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
| 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
−9.408346983209480964522234511396, −8.861913119283425787745533613033, −7.989140669701668047519131651502, −7.24119445195783161110628748242, −6.32325115003177354384208673052, −5.70539067440463519264988092545, −4.55751100186248914705850995356, −3.35964601021955354976089042882, −1.51345725959082442802938244596, −0.915635103977513096798626472525,
0.915635103977513096798626472525, 1.51345725959082442802938244596, 3.35964601021955354976089042882, 4.55751100186248914705850995356, 5.70539067440463519264988092545, 6.32325115003177354384208673052, 7.24119445195783161110628748242, 7.989140669701668047519131651502, 8.861913119283425787745533613033, 9.408346983209480964522234511396
