| L(s) = 1 | − 2-s + 4-s − 2·5-s − 2·7-s − 8-s + 2·10-s − 4·11-s − 13-s + 2·14-s + 16-s − 6·19-s − 2·20-s + 4·22-s + 4·23-s − 25-s + 26-s − 2·28-s − 8·29-s − 2·31-s − 32-s + 4·35-s + 6·37-s + 6·38-s + 2·40-s + 6·41-s − 8·43-s − 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.755·7-s − 0.353·8-s + 0.632·10-s − 1.20·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s − 1.37·19-s − 0.447·20-s + 0.852·22-s + 0.834·23-s − 1/5·25-s + 0.196·26-s − 0.377·28-s − 1.48·29-s − 0.359·31-s − 0.176·32-s + 0.676·35-s + 0.986·37-s + 0.973·38-s + 0.316·40-s + 0.937·41-s − 1.21·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−11.48471457324402872742962494510, −10.70725444808913514749917450090, −9.773699703719295926726447008045, −8.695739660347841822203445826112, −7.76516545555292037952264969735, −6.93707785293490008685942511031, −5.59584593532556316161391417712, −3.99394475360161838020150844401, −2.57523735650192716532994366231, 0,
2.57523735650192716532994366231, 3.99394475360161838020150844401, 5.59584593532556316161391417712, 6.93707785293490008685942511031, 7.76516545555292037952264969735, 8.695739660347841822203445826112, 9.773699703719295926726447008045, 10.70725444808913514749917450090, 11.48471457324402872742962494510
