| L(s) = 1 | − 1.87·2-s + 0.652·3-s + 1.53·4-s + 3.53·5-s − 1.22·6-s + 0.879·8-s − 2.57·9-s − 6.63·10-s + 12-s − 4.41·13-s + 2.30·15-s − 4.71·16-s + 5.24·17-s + 4.83·18-s − 1.81·19-s + 5.41·20-s − 6.33·23-s + 0.573·24-s + 7.47·25-s + 8.29·26-s − 3.63·27-s − 1.92·29-s − 4.33·30-s + 1.46·31-s + 7.10·32-s − 9.86·34-s − 3.94·36-s + ⋯ |
| L(s) = 1 | − 1.32·2-s + 0.376·3-s + 0.766·4-s + 1.57·5-s − 0.500·6-s + 0.310·8-s − 0.857·9-s − 2.09·10-s + 0.288·12-s − 1.22·13-s + 0.595·15-s − 1.17·16-s + 1.27·17-s + 1.14·18-s − 0.416·19-s + 1.21·20-s − 1.32·23-s + 0.117·24-s + 1.49·25-s + 1.62·26-s − 0.700·27-s − 0.356·29-s − 0.791·30-s + 0.263·31-s + 1.25·32-s − 1.69·34-s − 0.657·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.195368221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.195368221\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.87T + 2T^{2} \) |
| 3 | \( 1 - 0.652T + 3T^{2} \) |
| 5 | \( 1 - 3.53T + 5T^{2} \) |
| 13 | \( 1 + 4.41T + 13T^{2} \) |
| 17 | \( 1 - 5.24T + 17T^{2} \) |
| 19 | \( 1 + 1.81T + 19T^{2} \) |
| 23 | \( 1 + 6.33T + 23T^{2} \) |
| 29 | \( 1 + 1.92T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 - 4.45T + 37T^{2} \) |
| 41 | \( 1 + 0.283T + 41T^{2} \) |
| 43 | \( 1 - 3.41T + 43T^{2} \) |
| 47 | \( 1 + 4.55T + 47T^{2} \) |
| 53 | \( 1 + 7.23T + 53T^{2} \) |
| 59 | \( 1 - 9.53T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 + 0.694T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 + 2.34T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
| 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
−8.079835112255427124009097962817, −7.77356511780602651213909592745, −6.79943295253231906256909164071, −6.03037587997585309496377109195, −5.43895345034856127105266532632, −4.59941758818199410289292186516, −3.29816500528366063744911202911, −2.26517481599899415457560392687, −1.95838985996695562947404658157, −0.67674583322847028935246894540,
0.67674583322847028935246894540, 1.95838985996695562947404658157, 2.26517481599899415457560392687, 3.29816500528366063744911202911, 4.59941758818199410289292186516, 5.43895345034856127105266532632, 6.03037587997585309496377109195, 6.79943295253231906256909164071, 7.77356511780602651213909592745, 8.079835112255427124009097962817
