| L(s) = 1 | + 2-s + 3-s + 4-s − 0.473·5-s + 6-s + 8-s + 9-s − 0.473·10-s + 11-s + 12-s + 7.03·13-s − 0.473·15-s + 16-s − 4.64·17-s + 18-s + 6.55·19-s − 0.473·20-s + 22-s + 1.49·23-s + 24-s − 4.77·25-s + 7.03·26-s + 27-s − 2.11·29-s − 0.473·30-s + 6.83·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.211·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.149·10-s + 0.301·11-s + 0.288·12-s + 1.95·13-s − 0.122·15-s + 0.250·16-s − 1.12·17-s + 0.235·18-s + 1.50·19-s − 0.105·20-s + 0.213·22-s + 0.312·23-s + 0.204·24-s − 0.955·25-s + 1.37·26-s + 0.192·27-s − 0.393·29-s − 0.0864·30-s + 1.22·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.025885704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.025885704\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 0.473T + 5T^{2} \) |
| 13 | \( 1 - 7.03T + 13T^{2} \) |
| 17 | \( 1 + 4.64T + 17T^{2} \) |
| 19 | \( 1 - 6.55T + 19T^{2} \) |
| 23 | \( 1 - 1.49T + 23T^{2} \) |
| 29 | \( 1 + 2.11T + 29T^{2} \) |
| 31 | \( 1 - 6.83T + 31T^{2} \) |
| 37 | \( 1 + 4.32T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 0.158T + 43T^{2} \) |
| 47 | \( 1 - 0.270T + 47T^{2} \) |
| 53 | \( 1 + 9.66T + 53T^{2} \) |
| 59 | \( 1 - 9.82T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 + 1.21T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 + 6.13T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 7.54T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
| 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
−8.494215528139326457241087338656, −8.035108960098688965356727757114, −6.97031046973761023629117658104, −6.47528099241912813830761436166, −5.57499826582321434904289868805, −4.71514019293730435654437118023, −3.68805821982963134523050720605, −3.43103964868851183844983585810, −2.17482230841449785947130695846, −1.15420903325170302096098323378,
1.15420903325170302096098323378, 2.17482230841449785947130695846, 3.43103964868851183844983585810, 3.68805821982963134523050720605, 4.71514019293730435654437118023, 5.57499826582321434904289868805, 6.47528099241912813830761436166, 6.97031046973761023629117658104, 8.035108960098688965356727757114, 8.494215528139326457241087338656
