| L(s) = 1 | + 2-s + 3.01·3-s + 4-s + 1.87·5-s + 3.01·6-s + 3.72·7-s + 8-s + 6.10·9-s + 1.87·10-s + 3.39·11-s + 3.01·12-s − 6.93·13-s + 3.72·14-s + 5.65·15-s + 16-s − 5.87·17-s + 6.10·18-s − 3.22·19-s + 1.87·20-s + 11.2·21-s + 3.39·22-s − 7.20·23-s + 3.01·24-s − 1.48·25-s − 6.93·26-s + 9.38·27-s + 3.72·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.74·3-s + 0.5·4-s + 0.838·5-s + 1.23·6-s + 1.40·7-s + 0.353·8-s + 2.03·9-s + 0.592·10-s + 1.02·11-s + 0.871·12-s − 1.92·13-s + 0.994·14-s + 1.46·15-s + 0.250·16-s − 1.42·17-s + 1.44·18-s − 0.740·19-s + 0.419·20-s + 2.45·21-s + 0.723·22-s − 1.50·23-s + 0.616·24-s − 0.297·25-s − 1.35·26-s + 1.80·27-s + 0.703·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.393537607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.393537607\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 - 3.01T + 3T^{2} \) |
| 5 | \( 1 - 1.87T + 5T^{2} \) |
| 7 | \( 1 - 3.72T + 7T^{2} \) |
| 11 | \( 1 - 3.39T + 11T^{2} \) |
| 13 | \( 1 + 6.93T + 13T^{2} \) |
| 17 | \( 1 + 5.87T + 17T^{2} \) |
| 19 | \( 1 + 3.22T + 19T^{2} \) |
| 23 | \( 1 + 7.20T + 23T^{2} \) |
| 29 | \( 1 - 9.87T + 29T^{2} \) |
| 31 | \( 1 - 3.62T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 - 9.49T + 41T^{2} \) |
| 43 | \( 1 - 8.19T + 43T^{2} \) |
| 47 | \( 1 + 3.72T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 1.65T + 59T^{2} \) |
| 61 | \( 1 - 5.61T + 61T^{2} \) |
| 67 | \( 1 - 3.79T + 67T^{2} \) |
| 71 | \( 1 - 3.31T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 1.70T + 79T^{2} \) |
| 83 | \( 1 + 3.78T + 83T^{2} \) |
| 89 | \( 1 + 5.83T + 89T^{2} \) |
| 97 | \( 1 + 7.97T + 97T^{2} \) |
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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
−8.296368883412562491213686939611, −7.919506508757299699471386148043, −6.96713279880829924300001462861, −6.42076012266833970715004313858, −5.17774644788958653957596710498, −4.41140357809167339111712651102, −4.06945187619165614065951738550, −2.64804744604373811296206024588, −2.21154312405948191743009301715, −1.63687195966785061869825053516,
1.63687195966785061869825053516, 2.21154312405948191743009301715, 2.64804744604373811296206024588, 4.06945187619165614065951738550, 4.41140357809167339111712651102, 5.17774644788958653957596710498, 6.42076012266833970715004313858, 6.96713279880829924300001462861, 7.919506508757299699471386148043, 8.296368883412562491213686939611
