| L(s) = 1 | − 2-s + 0.204·3-s + 4-s − 5-s − 0.204·6-s + 0.0646·7-s − 8-s − 2.95·9-s + 10-s + 0.632·11-s + 0.204·12-s − 4.49·13-s − 0.0646·14-s − 0.204·15-s + 16-s − 7.35·17-s + 2.95·18-s − 2.97·19-s − 20-s + 0.0132·21-s − 0.632·22-s − 1.30·23-s − 0.204·24-s + 25-s + 4.49·26-s − 1.21·27-s + 0.0646·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.118·3-s + 0.5·4-s − 0.447·5-s − 0.0835·6-s + 0.0244·7-s − 0.353·8-s − 0.986·9-s + 0.316·10-s + 0.190·11-s + 0.0591·12-s − 1.24·13-s − 0.0172·14-s − 0.0528·15-s + 0.250·16-s − 1.78·17-s + 0.697·18-s − 0.681·19-s − 0.223·20-s + 0.00288·21-s − 0.134·22-s − 0.271·23-s − 0.0417·24-s + 0.200·25-s + 0.881·26-s − 0.234·27-s + 0.0122·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5658692351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5658692351\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 0.204T + 3T^{2} \) |
| 7 | \( 1 - 0.0646T + 7T^{2} \) |
| 11 | \( 1 - 0.632T + 11T^{2} \) |
| 13 | \( 1 + 4.49T + 13T^{2} \) |
| 17 | \( 1 + 7.35T + 17T^{2} \) |
| 19 | \( 1 + 2.97T + 19T^{2} \) |
| 23 | \( 1 + 1.30T + 23T^{2} \) |
| 29 | \( 1 + 4.04T + 29T^{2} \) |
| 31 | \( 1 + 9.86T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 7.78T + 43T^{2} \) |
| 47 | \( 1 - 8.52T + 47T^{2} \) |
| 53 | \( 1 - 5.61T + 53T^{2} \) |
| 59 | \( 1 + 5.26T + 59T^{2} \) |
| 61 | \( 1 - 2.26T + 61T^{2} \) |
| 67 | \( 1 - 7.55T + 67T^{2} \) |
| 71 | \( 1 - 6.36T + 71T^{2} \) |
| 73 | \( 1 + 5.56T + 73T^{2} \) |
| 79 | \( 1 - 4.15T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 0.448T + 89T^{2} \) |
| 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
−8.559209620526238449692350841181, −7.66722807249911838110101023302, −7.27521681553205414873534056421, −6.31473761256787893654745361803, −5.65646493269651895381037625702, −4.57226910495414661037456224087, −3.85852021637764210967936973862, −2.57934541710557770271428177066, −2.17063184291260954902643579330, −0.44276695751932409849268776283,
0.44276695751932409849268776283, 2.17063184291260954902643579330, 2.57934541710557770271428177066, 3.85852021637764210967936973862, 4.57226910495414661037456224087, 5.65646493269651895381037625702, 6.31473761256787893654745361803, 7.27521681553205414873534056421, 7.66722807249911838110101023302, 8.559209620526238449692350841181
