| L(s) = 1 | + 2-s + 0.409·3-s + 4-s − 2.52·5-s + 0.409·6-s + 2.33·7-s + 8-s − 2.83·9-s − 2.52·10-s − 4.50·11-s + 0.409·12-s − 1.19·13-s + 2.33·14-s − 1.03·15-s + 16-s + 5.67·17-s − 2.83·18-s + 2.84·19-s − 2.52·20-s + 0.955·21-s − 4.50·22-s + 5.65·23-s + 0.409·24-s + 1.38·25-s − 1.19·26-s − 2.38·27-s + 2.33·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.236·3-s + 0.5·4-s − 1.12·5-s + 0.167·6-s + 0.881·7-s + 0.353·8-s − 0.944·9-s − 0.798·10-s − 1.35·11-s + 0.118·12-s − 0.332·13-s + 0.623·14-s − 0.267·15-s + 0.250·16-s + 1.37·17-s − 0.667·18-s + 0.651·19-s − 0.564·20-s + 0.208·21-s − 0.960·22-s + 1.17·23-s + 0.0835·24-s + 0.276·25-s − 0.235·26-s − 0.459·27-s + 0.440·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.504797030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.504797030\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 0.409T + 3T^{2} \) |
| 5 | \( 1 + 2.52T + 5T^{2} \) |
| 7 | \( 1 - 2.33T + 7T^{2} \) |
| 11 | \( 1 + 4.50T + 11T^{2} \) |
| 13 | \( 1 + 1.19T + 13T^{2} \) |
| 17 | \( 1 - 5.67T + 17T^{2} \) |
| 19 | \( 1 - 2.84T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 3.21T + 29T^{2} \) |
| 31 | \( 1 + 1.11T + 31T^{2} \) |
| 37 | \( 1 - 3.80T + 37T^{2} \) |
| 41 | \( 1 - 7.93T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 0.253T + 47T^{2} \) |
| 53 | \( 1 - 1.03T + 53T^{2} \) |
| 59 | \( 1 + 2.05T + 59T^{2} \) |
| 61 | \( 1 - 0.761T + 61T^{2} \) |
| 67 | \( 1 - 5.45T + 67T^{2} \) |
| 71 | \( 1 - 7.46T + 71T^{2} \) |
| 73 | \( 1 - 5.90T + 73T^{2} \) |
| 79 | \( 1 - 9.64T + 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 - 5.83T + 89T^{2} \) |
| 97 | \( 1 - 5.55T + 97T^{2} \) |
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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
−8.060678055811550471782566953092, −7.71400980521899993516887516396, −7.33935046686084164265608991848, −5.94608711071103894193613724935, −5.30523840497441476683728736724, −4.79529437328332003903928056225, −3.76981356478398100583467335555, −3.06759972175369449779283540117, −2.33304998151854946541846381862, −0.795211765893054416550972199388,
0.795211765893054416550972199388, 2.33304998151854946541846381862, 3.06759972175369449779283540117, 3.76981356478398100583467335555, 4.79529437328332003903928056225, 5.30523840497441476683728736724, 5.94608711071103894193613724935, 7.33935046686084164265608991848, 7.71400980521899993516887516396, 8.060678055811550471782566953092
