L(s) = 1 | − 1.89·2-s + 1.34·3-s + 1.57·4-s + 4.23·5-s − 2.54·6-s − 3.43·7-s + 0.794·8-s − 1.18·9-s − 8.00·10-s + 5.77·11-s + 2.12·12-s + 1.41·13-s + 6.50·14-s + 5.70·15-s − 4.66·16-s + 5.54·17-s + 2.23·18-s − 2.00·19-s + 6.68·20-s − 4.63·21-s − 10.9·22-s − 3.79·23-s + 1.07·24-s + 12.9·25-s − 2.67·26-s − 5.63·27-s − 5.42·28-s + ⋯ |
L(s) = 1 | − 1.33·2-s + 0.778·3-s + 0.789·4-s + 1.89·5-s − 1.04·6-s − 1.29·7-s + 0.281·8-s − 0.394·9-s − 2.53·10-s + 1.73·11-s + 0.614·12-s + 0.391·13-s + 1.73·14-s + 1.47·15-s − 1.16·16-s + 1.34·17-s + 0.527·18-s − 0.460·19-s + 1.49·20-s − 1.01·21-s − 2.32·22-s − 0.792·23-s + 0.218·24-s + 2.58·25-s − 0.523·26-s − 1.08·27-s − 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.194019940\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.194019940\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 503 | \( 1 - T \) |
good | 2 | \( 1 + 1.89T + 2T^{2} \) |
| 3 | \( 1 - 1.34T + 3T^{2} \) |
| 5 | \( 1 - 4.23T + 5T^{2} \) |
| 7 | \( 1 + 3.43T + 7T^{2} \) |
| 11 | \( 1 - 5.77T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 - 5.54T + 17T^{2} \) |
| 19 | \( 1 + 2.00T + 19T^{2} \) |
| 23 | \( 1 + 3.79T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 - 4.53T + 31T^{2} \) |
| 37 | \( 1 - 4.04T + 37T^{2} \) |
| 41 | \( 1 - 6.27T + 41T^{2} \) |
| 43 | \( 1 + 0.618T + 43T^{2} \) |
| 47 | \( 1 - 4.16T + 47T^{2} \) |
| 53 | \( 1 - 6.53T + 53T^{2} \) |
| 59 | \( 1 - 6.82T + 59T^{2} \) |
| 61 | \( 1 + 5.36T + 61T^{2} \) |
| 67 | \( 1 - 0.00814T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 0.481T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 5.05T + 83T^{2} \) |
| 89 | \( 1 - 4.64T + 89T^{2} \) |
| 97 | \( 1 - 3.07T + 97T^{2} \) |
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
−10.25770171365423604602878611724, −9.774838795067279355208670839997, −9.156238838300678167057017005196, −8.810725163370550617562295426821, −7.47292836448779234544990122573, −6.33209981199964438883211585670, −5.85085328247624473701005801188, −3.77404740179310321805743194598, −2.47249056403073583645133424581, −1.33770366547113799831762316796,
1.33770366547113799831762316796, 2.47249056403073583645133424581, 3.77404740179310321805743194598, 5.85085328247624473701005801188, 6.33209981199964438883211585670, 7.47292836448779234544990122573, 8.810725163370550617562295426821, 9.156238838300678167057017005196, 9.774838795067279355208670839997, 10.25770171365423604602878611724