| L(s) = 1 | + 3·3-s − 22·7-s + 9·9-s + 14·11-s − 30·13-s + 62·17-s + 120·19-s − 66·21-s − 188·23-s + 27·27-s + 96·29-s − 184·31-s + 42·33-s + 406·37-s − 90·39-s + 130·41-s − 148·43-s − 448·47-s + 141·49-s + 186·51-s − 414·53-s + 360·57-s − 266·59-s − 838·61-s − 198·63-s − 248·67-s − 564·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.18·7-s + 1/3·9-s + 0.383·11-s − 0.640·13-s + 0.884·17-s + 1.44·19-s − 0.685·21-s − 1.70·23-s + 0.192·27-s + 0.614·29-s − 1.06·31-s + 0.221·33-s + 1.80·37-s − 0.369·39-s + 0.495·41-s − 0.524·43-s − 1.39·47-s + 0.411·49-s + 0.510·51-s − 1.07·53-s + 0.836·57-s − 0.586·59-s − 1.75·61-s − 0.395·63-s − 0.452·67-s − 0.984·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 22 T + p^{3} T^{2} \) |
| 11 | \( 1 - 14 T + p^{3} T^{2} \) |
| 13 | \( 1 + 30 T + p^{3} T^{2} \) |
| 17 | \( 1 - 62 T + p^{3} T^{2} \) |
| 19 | \( 1 - 120 T + p^{3} T^{2} \) |
| 23 | \( 1 + 188 T + p^{3} T^{2} \) |
| 29 | \( 1 - 96 T + p^{3} T^{2} \) |
| 31 | \( 1 + 184 T + p^{3} T^{2} \) |
| 37 | \( 1 - 406 T + p^{3} T^{2} \) |
| 41 | \( 1 - 130 T + p^{3} T^{2} \) |
| 43 | \( 1 + 148 T + p^{3} T^{2} \) |
| 47 | \( 1 + 448 T + p^{3} T^{2} \) |
| 53 | \( 1 + 414 T + p^{3} T^{2} \) |
| 59 | \( 1 + 266 T + p^{3} T^{2} \) |
| 61 | \( 1 + 838 T + p^{3} T^{2} \) |
| 67 | \( 1 + 248 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1020 T + p^{3} T^{2} \) |
| 73 | \( 1 - 484 T + p^{3} T^{2} \) |
| 79 | \( 1 - 48 T + p^{3} T^{2} \) |
| 83 | \( 1 + 548 T + p^{3} T^{2} \) |
| 89 | \( 1 + 650 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1816 T + p^{3} T^{2} \) |
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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
−9.247794658427247464572380289111, −7.983211813562633005714662518284, −7.49096000718456880484797243610, −6.45083210081252916111642344490, −5.71891009005085529735276490484, −4.49652751510173571581924428652, −3.45097327545773592520702094243, −2.82678559794350253139420087584, −1.45210364789876952006639097716, 0,
1.45210364789876952006639097716, 2.82678559794350253139420087584, 3.45097327545773592520702094243, 4.49652751510173571581924428652, 5.71891009005085529735276490484, 6.45083210081252916111642344490, 7.49096000718456880484797243610, 7.983211813562633005714662518284, 9.247794658427247464572380289111
