| L(s) = 1 | + 0.254·2-s − 1.25·3-s − 1.93·4-s + 0.681·5-s − 0.318·6-s − 0.745·7-s − 8-s − 1.42·9-s + 0.173·10-s − 3·11-s + 2.42·12-s − 4.87·13-s − 0.189·14-s − 0.854·15-s + 3.61·16-s − 3.06·17-s − 0.362·18-s − 1.93·19-s − 1.31·20-s + 0.935·21-s − 0.762·22-s + 8.18·23-s + 1.25·24-s − 4.53·25-s − 1.23·26-s + 5.55·27-s + 1.44·28-s + ⋯ |
| L(s) = 1 | + 0.179·2-s − 0.724·3-s − 0.967·4-s + 0.304·5-s − 0.130·6-s − 0.281·7-s − 0.353·8-s − 0.475·9-s + 0.0547·10-s − 0.904·11-s + 0.700·12-s − 1.35·13-s − 0.0506·14-s − 0.220·15-s + 0.904·16-s − 0.743·17-s − 0.0854·18-s − 0.444·19-s − 0.294·20-s + 0.204·21-s − 0.162·22-s + 1.70·23-s + 0.255·24-s − 0.907·25-s − 0.242·26-s + 1.06·27-s + 0.272·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 211 | \( 1 + T \) |
| good | 2 | \( 1 - 0.254T + 2T^{2} \) |
| 3 | \( 1 + 1.25T + 3T^{2} \) |
| 5 | \( 1 - 0.681T + 5T^{2} \) |
| 7 | \( 1 + 0.745T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 4.87T + 13T^{2} \) |
| 17 | \( 1 + 3.06T + 17T^{2} \) |
| 19 | \( 1 + 1.93T + 19T^{2} \) |
| 23 | \( 1 - 8.18T + 23T^{2} \) |
| 29 | \( 1 + 3.81T + 29T^{2} \) |
| 31 | \( 1 - 8.80T + 31T^{2} \) |
| 37 | \( 1 + 1.18T + 37T^{2} \) |
| 41 | \( 1 - 0.664T + 41T^{2} \) |
| 43 | \( 1 + 0.0164T + 43T^{2} \) |
| 47 | \( 1 + 3.42T + 47T^{2} \) |
| 53 | \( 1 + 2.93T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 7.04T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 - 0.745T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 - 3.62T + 83T^{2} \) |
| 89 | \( 1 + 6.96T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
| 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
−12.00821461084478954152924053222, −10.89371250454961810697414602759, −9.930365297428151232289134271127, −9.047376203865061538720041791407, −7.894885139221445945800743074749, −6.48943161012551701921878362895, −5.34187003293274512233148271326, −4.66440035597575773578198298393, −2.83809590489464985595579617332, 0,
2.83809590489464985595579617332, 4.66440035597575773578198298393, 5.34187003293274512233148271326, 6.48943161012551701921878362895, 7.894885139221445945800743074749, 9.047376203865061538720041791407, 9.930365297428151232289134271127, 10.89371250454961810697414602759, 12.00821461084478954152924053222
