| L(s) = 1 | − 1.53·2-s + 0.0485·3-s + 0.360·4-s − 0.584·5-s − 0.0746·6-s − 1.23·7-s + 2.51·8-s − 2.99·9-s + 0.897·10-s + 11-s + 0.0175·12-s − 0.643·13-s + 1.89·14-s − 0.0283·15-s − 4.59·16-s − 17-s + 4.60·18-s − 5.42·19-s − 0.210·20-s − 0.0599·21-s − 1.53·22-s − 9.24·23-s + 0.122·24-s − 4.65·25-s + 0.988·26-s − 0.291·27-s − 0.445·28-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 0.0280·3-s + 0.180·4-s − 0.261·5-s − 0.0304·6-s − 0.466·7-s + 0.890·8-s − 0.999·9-s + 0.283·10-s + 0.301·11-s + 0.00506·12-s − 0.178·13-s + 0.506·14-s − 0.00732·15-s − 1.14·16-s − 0.242·17-s + 1.08·18-s − 1.24·19-s − 0.0471·20-s − 0.0130·21-s − 0.327·22-s − 1.92·23-s + 0.0249·24-s − 0.931·25-s + 0.193·26-s − 0.0560·27-s − 0.0842·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02007976054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02007976054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 3 | \( 1 - 0.0485T + 3T^{2} \) |
| 5 | \( 1 + 0.584T + 5T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 13 | \( 1 + 0.643T + 13T^{2} \) |
| 19 | \( 1 + 5.42T + 19T^{2} \) |
| 23 | \( 1 + 9.24T + 23T^{2} \) |
| 29 | \( 1 + 8.11T + 29T^{2} \) |
| 31 | \( 1 + 6.20T + 31T^{2} \) |
| 37 | \( 1 - 5.04T + 37T^{2} \) |
| 41 | \( 1 + 4.44T + 41T^{2} \) |
| 47 | \( 1 + 7.15T + 47T^{2} \) |
| 53 | \( 1 - 9.14T + 53T^{2} \) |
| 59 | \( 1 - 0.525T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 0.835T + 67T^{2} \) |
| 71 | \( 1 - 0.263T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 0.175T + 79T^{2} \) |
| 83 | \( 1 - 7.40T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 1.48T + 97T^{2} \) |
| 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
−7.976949122884250663856265036353, −7.44841667231706700954002083593, −6.48372456352895292819361873703, −5.97026222911011922691960047962, −5.08998608522367201272439582850, −4.06526310317043381015105695414, −3.64482807352029559669615615102, −2.33364123519154206057459041091, −1.72935328863177148595637251189, −0.079236661994906028984778591869,
0.079236661994906028984778591869, 1.72935328863177148595637251189, 2.33364123519154206057459041091, 3.64482807352029559669615615102, 4.06526310317043381015105695414, 5.08998608522367201272439582850, 5.97026222911011922691960047962, 6.48372456352895292819361873703, 7.44841667231706700954002083593, 7.976949122884250663856265036353
