| L(s) = 1 | + 2-s + 3-s + 4-s − 3.03·5-s + 6-s − 0.874·7-s + 8-s + 9-s − 3.03·10-s + 6.29·11-s + 12-s + 1.49·13-s − 0.874·14-s − 3.03·15-s + 16-s − 6.72·17-s + 18-s − 3.03·20-s − 0.874·21-s + 6.29·22-s + 4·23-s + 24-s + 4.20·25-s + 1.49·26-s + 27-s − 0.874·28-s + 8.50·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.35·5-s + 0.408·6-s − 0.330·7-s + 0.353·8-s + 0.333·9-s − 0.959·10-s + 1.89·11-s + 0.288·12-s + 0.414·13-s − 0.233·14-s − 0.783·15-s + 0.250·16-s − 1.63·17-s + 0.235·18-s − 0.678·20-s − 0.190·21-s + 1.34·22-s + 0.834·23-s + 0.204·24-s + 0.840·25-s + 0.292·26-s + 0.192·27-s − 0.165·28-s + 1.57·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.928700770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.928700770\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 3.03T + 5T^{2} \) |
| 7 | \( 1 + 0.874T + 7T^{2} \) |
| 11 | \( 1 - 6.29T + 11T^{2} \) |
| 13 | \( 1 - 1.49T + 13T^{2} \) |
| 17 | \( 1 + 6.72T + 17T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 8.50T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 - 3.14T + 37T^{2} \) |
| 41 | \( 1 - 9.39T + 41T^{2} \) |
| 43 | \( 1 - 9.30T + 43T^{2} \) |
| 47 | \( 1 + 2.83T + 47T^{2} \) |
| 53 | \( 1 - 1.21T + 53T^{2} \) |
| 59 | \( 1 - 8.76T + 59T^{2} \) |
| 61 | \( 1 - 9.49T + 61T^{2} \) |
| 67 | \( 1 - 2.47T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 8.01T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 - 3.70T + 83T^{2} \) |
| 89 | \( 1 - 2.92T + 89T^{2} \) |
| 97 | \( 1 + 8.97T + 97T^{2} \) |
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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
−8.879382052377477236860891914629, −8.402011588150242680515929305838, −7.32152395712118529631706348331, −6.78405413659111973111347867777, −6.12060479724727020114896589092, −4.59441844604068582629868186995, −4.14018616142628493528406970434, −3.51453128258906925215730234972, −2.50473425268954462032038841031, −1.03635862871435097863660322993,
1.03635862871435097863660322993, 2.50473425268954462032038841031, 3.51453128258906925215730234972, 4.14018616142628493528406970434, 4.59441844604068582629868186995, 6.12060479724727020114896589092, 6.78405413659111973111347867777, 7.32152395712118529631706348331, 8.402011588150242680515929305838, 8.879382052377477236860891914629
