| L(s) = 1 | − 2.22·2-s + 0.200·3-s + 2.95·4-s + 3.39·5-s − 0.446·6-s − 0.867·7-s − 2.13·8-s − 2.95·9-s − 7.56·10-s − 11-s + 0.593·12-s − 5.52·13-s + 1.93·14-s + 0.682·15-s − 1.16·16-s + 3.54·17-s + 6.59·18-s + 0.206·19-s + 10.0·20-s − 0.174·21-s + 2.22·22-s + 6.80·23-s − 0.428·24-s + 6.55·25-s + 12.3·26-s − 1.19·27-s − 2.56·28-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 0.115·3-s + 1.47·4-s + 1.52·5-s − 0.182·6-s − 0.327·7-s − 0.754·8-s − 0.986·9-s − 2.39·10-s − 0.301·11-s + 0.171·12-s − 1.53·13-s + 0.516·14-s + 0.176·15-s − 0.291·16-s + 0.858·17-s + 1.55·18-s + 0.0474·19-s + 2.24·20-s − 0.0379·21-s + 0.474·22-s + 1.41·23-s − 0.0874·24-s + 1.31·25-s + 2.41·26-s − 0.230·27-s − 0.485·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9590710103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9590710103\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 547 | \( 1 - T \) |
| good | 2 | \( 1 + 2.22T + 2T^{2} \) |
| 3 | \( 1 - 0.200T + 3T^{2} \) |
| 5 | \( 1 - 3.39T + 5T^{2} \) |
| 7 | \( 1 + 0.867T + 7T^{2} \) |
| 13 | \( 1 + 5.52T + 13T^{2} \) |
| 17 | \( 1 - 3.54T + 17T^{2} \) |
| 19 | \( 1 - 0.206T + 19T^{2} \) |
| 23 | \( 1 - 6.80T + 23T^{2} \) |
| 29 | \( 1 - 3.50T + 29T^{2} \) |
| 31 | \( 1 - 3.97T + 31T^{2} \) |
| 37 | \( 1 - 5.88T + 37T^{2} \) |
| 41 | \( 1 + 0.545T + 41T^{2} \) |
| 43 | \( 1 + 9.67T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 4.89T + 53T^{2} \) |
| 59 | \( 1 - 6.94T + 59T^{2} \) |
| 61 | \( 1 - 0.613T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 2.13T + 71T^{2} \) |
| 73 | \( 1 + 0.891T + 73T^{2} \) |
| 79 | \( 1 + 8.08T + 79T^{2} \) |
| 83 | \( 1 + 0.211T + 83T^{2} \) |
| 89 | \( 1 + 6.97T + 89T^{2} \) |
| 97 | \( 1 + 3.76T + 97T^{2} \) |
| 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
−8.244229058438457469945556209408, −7.51038669682747575104843301415, −6.84282919439337029488987812898, −6.14197869931260612372542576819, −5.38515010186369237862567773924, −4.73683701391399306605386560968, −2.90819798569547708021816011098, −2.66692665552306693833765071628, −1.68883830203494825392080864547, −0.64690885259017827709819175450,
0.64690885259017827709819175450, 1.68883830203494825392080864547, 2.66692665552306693833765071628, 2.90819798569547708021816011098, 4.73683701391399306605386560968, 5.38515010186369237862567773924, 6.14197869931260612372542576819, 6.84282919439337029488987812898, 7.51038669682747575104843301415, 8.244229058438457469945556209408
