| L(s) = 1 | + 1.56·2-s + 3-s + 0.438·4-s − 5-s + 1.56·6-s − 2.43·8-s + 9-s − 1.56·10-s + 11-s + 0.438·12-s + 7.12·13-s − 15-s − 4.68·16-s − 0.561·17-s + 1.56·18-s + 2.56·19-s − 0.438·20-s + 1.56·22-s − 1.43·23-s − 2.43·24-s + 25-s + 11.1·26-s + 27-s + 1.68·29-s − 1.56·30-s + 5.12·31-s − 2.43·32-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.577·3-s + 0.219·4-s − 0.447·5-s + 0.637·6-s − 0.862·8-s + 0.333·9-s − 0.493·10-s + 0.301·11-s + 0.126·12-s + 1.97·13-s − 0.258·15-s − 1.17·16-s − 0.136·17-s + 0.368·18-s + 0.587·19-s − 0.0980·20-s + 0.332·22-s − 0.299·23-s − 0.497·24-s + 0.200·25-s + 2.18·26-s + 0.192·27-s + 0.312·29-s − 0.285·30-s + 0.920·31-s − 0.431·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.185002023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.185002023\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 13 | \( 1 - 7.12T + 13T^{2} \) |
| 17 | \( 1 + 0.561T + 17T^{2} \) |
| 19 | \( 1 - 2.56T + 19T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 - 1.68T + 29T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 2.56T + 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 3.43T + 61T^{2} \) |
| 67 | \( 1 - 6.87T + 67T^{2} \) |
| 71 | \( 1 - 5.12T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 5.43T + 83T^{2} \) |
| 89 | \( 1 - 2.31T + 89T^{2} \) |
| 97 | \( 1 + 2.80T + 97T^{2} \) |
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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
−7.932373054910410354904621145518, −6.93080235748203525682620372629, −6.31162947447342719203431230252, −5.74536878174168428760975050260, −4.80263934512721989967037146838, −4.23437256111484461664013150973, −3.40323270163961522059626038354, −3.25131561771431775284905532839, −1.97275898919609058323776240582, −0.854623194947728614831023735295,
0.854623194947728614831023735295, 1.97275898919609058323776240582, 3.25131561771431775284905532839, 3.40323270163961522059626038354, 4.23437256111484461664013150973, 4.80263934512721989967037146838, 5.74536878174168428760975050260, 6.31162947447342719203431230252, 6.93080235748203525682620372629, 7.932373054910410354904621145518
