| L(s) = 1 | − 4.22·2-s + 0.642·3-s + 9.83·4-s + 5·5-s − 2.71·6-s − 28.8·7-s − 7.76·8-s − 26.5·9-s − 21.1·10-s + 11·11-s + 6.32·12-s + 4.32·13-s + 121.·14-s + 3.21·15-s − 45.9·16-s + 31.1·17-s + 112.·18-s − 19·19-s + 49.1·20-s − 18.5·21-s − 46.4·22-s − 205.·23-s − 4.99·24-s + 25·25-s − 18.2·26-s − 34.4·27-s − 283.·28-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 0.123·3-s + 1.22·4-s + 0.447·5-s − 0.184·6-s − 1.55·7-s − 0.343·8-s − 0.984·9-s − 0.667·10-s + 0.301·11-s + 0.152·12-s + 0.0922·13-s + 2.32·14-s + 0.0553·15-s − 0.717·16-s + 0.444·17-s + 1.47·18-s − 0.229·19-s + 0.549·20-s − 0.192·21-s − 0.450·22-s − 1.85·23-s − 0.0424·24-s + 0.200·25-s − 0.137·26-s − 0.245·27-s − 1.91·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3627587956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3627587956\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 4.22T + 8T^{2} \) |
| 3 | \( 1 - 0.642T + 27T^{2} \) |
| 7 | \( 1 + 28.8T + 343T^{2} \) |
| 13 | \( 1 - 4.32T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.1T + 4.91e3T^{2} \) |
| 23 | \( 1 + 205.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 222.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 45.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 166.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 184.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 182.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 240.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 299.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 345.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 439.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 74.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 42.3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 980.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 840.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 125.T + 9.12e5T^{2} \) |
| 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
−9.523907866907210272499892502835, −8.920118729646659519660711149380, −8.085125893663902945379391477959, −7.23039900503775369512138293278, −6.25455820463929774016179266764, −5.73875165817227164542288149259, −3.97151841941071704892638515422, −2.88408451797116853526101906571, −1.85247638381794103858250924903, −0.37917059473055766238661040360,
0.37917059473055766238661040360, 1.85247638381794103858250924903, 2.88408451797116853526101906571, 3.97151841941071704892638515422, 5.73875165817227164542288149259, 6.25455820463929774016179266764, 7.23039900503775369512138293278, 8.085125893663902945379391477959, 8.920118729646659519660711149380, 9.523907866907210272499892502835
