| L(s) = 1 | + 2-s + 2.47·3-s + 4-s − 2.72·5-s + 2.47·6-s + 0.694·7-s + 8-s + 3.10·9-s − 2.72·10-s + 0.791·11-s + 2.47·12-s − 7.05·13-s + 0.694·14-s − 6.72·15-s + 16-s − 5.78·17-s + 3.10·18-s + 0.885·19-s − 2.72·20-s + 1.71·21-s + 0.791·22-s + 5.00·23-s + 2.47·24-s + 2.39·25-s − 7.05·26-s + 0.268·27-s + 0.694·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.42·3-s + 0.5·4-s − 1.21·5-s + 1.00·6-s + 0.262·7-s + 0.353·8-s + 1.03·9-s − 0.860·10-s + 0.238·11-s + 0.713·12-s − 1.95·13-s + 0.185·14-s − 1.73·15-s + 0.250·16-s − 1.40·17-s + 0.732·18-s + 0.203·19-s − 0.608·20-s + 0.374·21-s + 0.168·22-s + 1.04·23-s + 0.504·24-s + 0.479·25-s − 1.38·26-s + 0.0517·27-s + 0.131·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3001 | \( 1+O(T) \) |
| good | 3 | \( 1 - 2.47T + 3T^{2} \) |
| 5 | \( 1 + 2.72T + 5T^{2} \) |
| 7 | \( 1 - 0.694T + 7T^{2} \) |
| 11 | \( 1 - 0.791T + 11T^{2} \) |
| 13 | \( 1 + 7.05T + 13T^{2} \) |
| 17 | \( 1 + 5.78T + 17T^{2} \) |
| 19 | \( 1 - 0.885T + 19T^{2} \) |
| 23 | \( 1 - 5.00T + 23T^{2} \) |
| 29 | \( 1 + 6.26T + 29T^{2} \) |
| 31 | \( 1 - 4.12T + 31T^{2} \) |
| 37 | \( 1 + 5.54T + 37T^{2} \) |
| 41 | \( 1 + 6.49T + 41T^{2} \) |
| 43 | \( 1 + 2.58T + 43T^{2} \) |
| 47 | \( 1 - 3.78T + 47T^{2} \) |
| 53 | \( 1 + 8.49T + 53T^{2} \) |
| 59 | \( 1 - 0.649T + 59T^{2} \) |
| 61 | \( 1 - 8.42T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 3.22T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 6.66T + 79T^{2} \) |
| 83 | \( 1 - 1.55T + 83T^{2} \) |
| 89 | \( 1 + 2.10T + 89T^{2} \) |
| 97 | \( 1 + 1.35T + 97T^{2} \) |
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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
−7.70522787155078383826956095046, −7.16545245955161270028953892195, −6.62597481261943762977664225110, −5.17289580401180736106782470619, −4.68090640898503858909591643173, −3.95154663471859399124073624835, −3.26779655116838159355120199544, −2.56511913742479116575429111827, −1.81163109547347218704101321761, 0,
1.81163109547347218704101321761, 2.56511913742479116575429111827, 3.26779655116838159355120199544, 3.95154663471859399124073624835, 4.68090640898503858909591643173, 5.17289580401180736106782470619, 6.62597481261943762977664225110, 7.16545245955161270028953892195, 7.70522787155078383826956095046
