| L(s) = 1 | − 1.56·2-s + 3-s + 0.438·4-s − 1.56·6-s + 3.56·7-s + 2.43·8-s + 9-s − 2·11-s + 0.438·12-s + 13-s − 5.56·14-s − 4.68·16-s + 0.438·17-s − 1.56·18-s − 3.56·19-s + 3.56·21-s + 3.12·22-s + 5·23-s + 2.43·24-s − 1.56·26-s + 27-s + 1.56·28-s + 7.56·29-s − 5.12·31-s + 2.43·32-s − 2·33-s − 0.684·34-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.577·3-s + 0.219·4-s − 0.637·6-s + 1.34·7-s + 0.862·8-s + 0.333·9-s − 0.603·11-s + 0.126·12-s + 0.277·13-s − 1.48·14-s − 1.17·16-s + 0.106·17-s − 0.368·18-s − 0.817·19-s + 0.777·21-s + 0.665·22-s + 1.04·23-s + 0.497·24-s − 0.306·26-s + 0.192·27-s + 0.295·28-s + 1.40·29-s − 0.920·31-s + 0.431·32-s − 0.348·33-s − 0.117·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.449838138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.449838138\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 0.438T + 17T^{2} \) |
| 19 | \( 1 + 3.56T + 19T^{2} \) |
| 23 | \( 1 - 5T + 23T^{2} \) |
| 29 | \( 1 - 7.56T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 1.56T + 41T^{2} \) |
| 43 | \( 1 - 4.56T + 43T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 4.12T + 61T^{2} \) |
| 67 | \( 1 + 2.24T + 67T^{2} \) |
| 71 | \( 1 - T + 71T^{2} \) |
| 73 | \( 1 + 9.68T + 73T^{2} \) |
| 79 | \( 1 - 6.80T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 0.561T + 89T^{2} \) |
| 97 | \( 1 - 9.12T + 97T^{2} \) |
| 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
−8.669681397964011109207828684804, −7.994729862267382647879033717713, −7.48991097791017129955641206943, −6.70445482231482709108752657254, −5.42324713103043363572157156872, −4.71924881410014352227092953463, −3.99853965729781136897422570655, −2.66742138424656439524716914777, −1.80577840753791379799851723050, −0.862694910913244328608913778349,
0.862694910913244328608913778349, 1.80577840753791379799851723050, 2.66742138424656439524716914777, 3.99853965729781136897422570655, 4.71924881410014352227092953463, 5.42324713103043363572157156872, 6.70445482231482709108752657254, 7.48991097791017129955641206943, 7.994729862267382647879033717713, 8.669681397964011109207828684804
