| L(s) = 1 | + 2-s − 2.24·3-s + 4-s + 3.12·5-s − 2.24·6-s + 2.41·7-s + 8-s + 2.05·9-s + 3.12·10-s + 2.69·11-s − 2.24·12-s − 2.02·13-s + 2.41·14-s − 7.02·15-s + 16-s − 2.24·17-s + 2.05·18-s + 3.55·19-s + 3.12·20-s − 5.43·21-s + 2.69·22-s − 3.57·23-s − 2.24·24-s + 4.76·25-s − 2.02·26-s + 2.12·27-s + 2.41·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.29·3-s + 0.5·4-s + 1.39·5-s − 0.917·6-s + 0.913·7-s + 0.353·8-s + 0.685·9-s + 0.988·10-s + 0.813·11-s − 0.649·12-s − 0.561·13-s + 0.646·14-s − 1.81·15-s + 0.250·16-s − 0.544·17-s + 0.484·18-s + 0.816·19-s + 0.698·20-s − 1.18·21-s + 0.575·22-s − 0.745·23-s − 0.458·24-s + 0.953·25-s − 0.397·26-s + 0.408·27-s + 0.456·28-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
| \(L(1)\) |
\(\approx\) |
\(3.015931126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.015931126\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.24T + 3T^{2} \) |
| 5 | \( 1 - 3.12T + 5T^{2} \) |
| 7 | \( 1 - 2.41T + 7T^{2} \) |
| 11 | \( 1 - 2.69T + 11T^{2} \) |
| 13 | \( 1 + 2.02T + 13T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 - 3.55T + 19T^{2} \) |
| 23 | \( 1 + 3.57T + 23T^{2} \) |
| 29 | \( 1 - 4.18T + 29T^{2} \) |
| 31 | \( 1 - 1.43T + 31T^{2} \) |
| 37 | \( 1 - 7.27T + 37T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 - 0.636T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 + 0.271T + 53T^{2} \) |
| 59 | \( 1 + 3.56T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 2.88T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 1.79T + 79T^{2} \) |
| 83 | \( 1 - 1.15T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 + 3.18T + 97T^{2} \) |
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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
−8.415381928858538378091620209452, −7.39994166172194037703479029839, −6.64676936665255887743035926035, −5.94770626876438044906840491709, −5.64774203818576775317042624448, −4.75496431952679137509069824126, −4.33597460304281602705343425243, −2.83580678732649013707831753430, −1.90682822384750269254535852648, −1.02507627586870638068947261151,
1.02507627586870638068947261151, 1.90682822384750269254535852648, 2.83580678732649013707831753430, 4.33597460304281602705343425243, 4.75496431952679137509069824126, 5.64774203818576775317042624448, 5.94770626876438044906840491709, 6.64676936665255887743035926035, 7.39994166172194037703479029839, 8.415381928858538378091620209452
