L(s) = 1 | − 1.97·2-s − 0.571·3-s + 1.89·4-s − 1.23·5-s + 1.12·6-s + 2.55·7-s + 0.199·8-s − 2.67·9-s + 2.42·10-s + 11-s − 1.08·12-s − 5.05·14-s + 0.703·15-s − 4.19·16-s + 5.97·17-s + 5.27·18-s + 5.32·19-s − 2.33·20-s − 1.46·21-s − 1.97·22-s − 3.26·23-s − 0.113·24-s − 3.48·25-s + 3.24·27-s + 4.85·28-s − 2.86·29-s − 1.38·30-s + ⋯ |
L(s) = 1 | − 1.39·2-s − 0.329·3-s + 0.949·4-s − 0.550·5-s + 0.460·6-s + 0.966·7-s + 0.0704·8-s − 0.891·9-s + 0.768·10-s + 0.301·11-s − 0.313·12-s − 1.34·14-s + 0.181·15-s − 1.04·16-s + 1.44·17-s + 1.24·18-s + 1.22·19-s − 0.522·20-s − 0.319·21-s − 0.420·22-s − 0.680·23-s − 0.0232·24-s − 0.697·25-s + 0.623·27-s + 0.918·28-s − 0.531·29-s − 0.253·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6558410306\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6558410306\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.97T + 2T^{2} \) |
| 3 | \( 1 + 0.571T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 - 2.55T + 7T^{2} \) |
| 17 | \( 1 - 5.97T + 17T^{2} \) |
| 19 | \( 1 - 5.32T + 19T^{2} \) |
| 23 | \( 1 + 3.26T + 23T^{2} \) |
| 29 | \( 1 + 2.86T + 29T^{2} \) |
| 31 | \( 1 + 2.90T + 31T^{2} \) |
| 37 | \( 1 - 9.62T + 37T^{2} \) |
| 41 | \( 1 - 3.47T + 41T^{2} \) |
| 43 | \( 1 - 4.53T + 43T^{2} \) |
| 47 | \( 1 + 0.858T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 + 8.76T + 59T^{2} \) |
| 61 | \( 1 - 2.76T + 61T^{2} \) |
| 67 | \( 1 + 3.64T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 - 5.63T + 73T^{2} \) |
| 79 | \( 1 - 7.57T + 79T^{2} \) |
| 83 | \( 1 - 0.707T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 3.45T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.407194918673290464839797550726, −8.222786460206997006740969472733, −7.911069774962083041087146874011, −7.38527505002233823127076118184, −6.10756088533602689363667298770, −5.33660975450829493838282839091, −4.32039115993984026991395554841, −3.14869898770133128956153333974, −1.76500988075011739284750518644, −0.71288728839891128672135862686,
0.71288728839891128672135862686, 1.76500988075011739284750518644, 3.14869898770133128956153333974, 4.32039115993984026991395554841, 5.33660975450829493838282839091, 6.10756088533602689363667298770, 7.38527505002233823127076118184, 7.911069774962083041087146874011, 8.222786460206997006740969472733, 9.407194918673290464839797550726