| L(s) = 1 | + 2.36·2-s + 1.69·3-s + 3.58·4-s + 0.763·5-s + 4.01·6-s − 0.178·7-s + 3.74·8-s − 0.119·9-s + 1.80·10-s − 3.15·11-s + 6.08·12-s − 4.14·13-s − 0.420·14-s + 1.29·15-s + 1.67·16-s − 7.28·17-s − 0.281·18-s + 7.21·19-s + 2.73·20-s − 0.302·21-s − 7.44·22-s + 1.98·23-s + 6.35·24-s − 4.41·25-s − 9.79·26-s − 5.29·27-s − 0.637·28-s + ⋯ |
| L(s) = 1 | + 1.67·2-s + 0.979·3-s + 1.79·4-s + 0.341·5-s + 1.63·6-s − 0.0672·7-s + 1.32·8-s − 0.0397·9-s + 0.570·10-s − 0.949·11-s + 1.75·12-s − 1.14·13-s − 0.112·14-s + 0.334·15-s + 0.418·16-s − 1.76·17-s − 0.0664·18-s + 1.65·19-s + 0.612·20-s − 0.0659·21-s − 1.58·22-s + 0.414·23-s + 1.29·24-s − 0.883·25-s − 1.92·26-s − 1.01·27-s − 0.120·28-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
| \(L(1)\) |
\(\approx\) |
\(4.361050094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.361050094\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.36T + 2T^{2} \) |
| 3 | \( 1 - 1.69T + 3T^{2} \) |
| 5 | \( 1 - 0.763T + 5T^{2} \) |
| 7 | \( 1 + 0.178T + 7T^{2} \) |
| 11 | \( 1 + 3.15T + 11T^{2} \) |
| 13 | \( 1 + 4.14T + 13T^{2} \) |
| 17 | \( 1 + 7.28T + 17T^{2} \) |
| 19 | \( 1 - 7.21T + 19T^{2} \) |
| 23 | \( 1 - 1.98T + 23T^{2} \) |
| 29 | \( 1 - 5.44T + 29T^{2} \) |
| 31 | \( 1 - 9.61T + 31T^{2} \) |
| 37 | \( 1 - 8.98T + 37T^{2} \) |
| 41 | \( 1 - 1.85T + 41T^{2} \) |
| 43 | \( 1 - 8.91T + 43T^{2} \) |
| 47 | \( 1 - 0.850T + 47T^{2} \) |
| 53 | \( 1 - 6.82T + 53T^{2} \) |
| 59 | \( 1 - 7.34T + 59T^{2} \) |
| 61 | \( 1 + 3.45T + 61T^{2} \) |
| 67 | \( 1 - 8.65T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 3.24T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 + 6.24T + 83T^{2} \) |
| 89 | \( 1 - 1.86T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
| 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
−11.30407966840685624990241934889, −10.06173125536808152593361694227, −9.196366252121417120436539475024, −8.002160082276819077700399049891, −7.13277549701633152035017964682, −6.03178387676396212260467885065, −5.06920394790344044348690905308, −4.22815535936294159714973783382, −2.74341327368009217583143951937, −2.54634833333766832543812901430,
2.54634833333766832543812901430, 2.74341327368009217583143951937, 4.22815535936294159714973783382, 5.06920394790344044348690905308, 6.03178387676396212260467885065, 7.13277549701633152035017964682, 8.002160082276819077700399049891, 9.196366252121417120436539475024, 10.06173125536808152593361694227, 11.30407966840685624990241934889
