L(s) = 1 | + 2-s − 3-s + 4-s − 0.714·5-s − 6-s + 1.94·7-s + 8-s + 9-s − 0.714·10-s − 2.05·11-s − 12-s + 13-s + 1.94·14-s + 0.714·15-s + 16-s − 7.24·17-s + 18-s + 3.15·19-s − 0.714·20-s − 1.94·21-s − 2.05·22-s + 2.78·23-s − 24-s − 4.48·25-s + 26-s − 27-s + 1.94·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.319·5-s − 0.408·6-s + 0.734·7-s + 0.353·8-s + 0.333·9-s − 0.225·10-s − 0.618·11-s − 0.288·12-s + 0.277·13-s + 0.519·14-s + 0.184·15-s + 0.250·16-s − 1.75·17-s + 0.235·18-s + 0.723·19-s − 0.159·20-s − 0.424·21-s − 0.437·22-s + 0.581·23-s − 0.204·24-s − 0.897·25-s + 0.196·26-s − 0.192·27-s + 0.367·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.475794332\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.475794332\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 0.714T + 5T^{2} \) |
| 7 | \( 1 - 1.94T + 7T^{2} \) |
| 11 | \( 1 + 2.05T + 11T^{2} \) |
| 17 | \( 1 + 7.24T + 17T^{2} \) |
| 19 | \( 1 - 3.15T + 19T^{2} \) |
| 23 | \( 1 - 2.78T + 23T^{2} \) |
| 29 | \( 1 - 3.55T + 29T^{2} \) |
| 31 | \( 1 - 3.15T + 31T^{2} \) |
| 37 | \( 1 + 6.57T + 37T^{2} \) |
| 41 | \( 1 + 1.66T + 41T^{2} \) |
| 43 | \( 1 - 5.71T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 + 9.25T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 3.69T + 67T^{2} \) |
| 71 | \( 1 - 9.88T + 71T^{2} \) |
| 73 | \( 1 - 0.674T + 73T^{2} \) |
| 79 | \( 1 + 2.88T + 79T^{2} \) |
| 83 | \( 1 - 7.36T + 83T^{2} \) |
| 89 | \( 1 - 2.45T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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
−7.73486411488295325089686752372, −6.95569079233755459974873005947, −6.47580168297787108769672567135, −5.52495794326033552354842306381, −5.07341066998293257264293973467, −4.37995032433286431698614376393, −3.75124618500883148556111215544, −2.66090905839876371595334306645, −1.90854106772840480328941850875, −0.71533360484807405265475362636,
0.71533360484807405265475362636, 1.90854106772840480328941850875, 2.66090905839876371595334306645, 3.75124618500883148556111215544, 4.37995032433286431698614376393, 5.07341066998293257264293973467, 5.52495794326033552354842306381, 6.47580168297787108769672567135, 6.95569079233755459974873005947, 7.73486411488295325089686752372