| L(s) = 1 | − 3-s − 5-s + 9-s + 4·11-s + 13-s + 15-s + 6·17-s + 4·19-s + 25-s − 27-s + 2·29-s + 8·31-s − 4·33-s + 6·37-s − 39-s − 10·41-s − 4·43-s − 45-s + 8·47-s − 7·49-s − 6·51-s + 6·53-s − 4·55-s − 4·57-s − 12·59-s − 10·61-s − 65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.258·15-s + 1.45·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s + 0.986·37-s − 0.160·39-s − 1.56·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s − 49-s − 0.840·51-s + 0.824·53-s − 0.539·55-s − 0.529·57-s − 1.56·59-s − 1.28·61-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.858776337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.858776337\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.947024942201543620404002077820, −7.35636779219708892660325177777, −6.52431635247964933864233169765, −6.04040524925983557166863746843, −5.15610259074422422063688994184, −4.47680162456485182475114267127, −3.62284880782181867823314174285, −2.99583786789386571784697269860, −1.50703669572845863680434335287, −0.815992074978217174148687342212,
0.815992074978217174148687342212, 1.50703669572845863680434335287, 2.99583786789386571784697269860, 3.62284880782181867823314174285, 4.47680162456485182475114267127, 5.15610259074422422063688994184, 6.04040524925983557166863746843, 6.52431635247964933864233169765, 7.35636779219708892660325177777, 7.947024942201543620404002077820
