| L(s) = 1 | + 2-s + 4-s − 5-s − 3.37·7-s + 8-s − 10-s + 1.37·11-s + 1.37·13-s − 3.37·14-s + 16-s + 1.37·17-s − 1.37·19-s − 20-s + 1.37·22-s − 3.37·23-s + 25-s + 1.37·26-s − 3.37·28-s − 6·29-s + 2.74·31-s + 32-s + 1.37·34-s + 3.37·35-s − 37-s − 1.37·38-s − 40-s − 8.74·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.27·7-s + 0.353·8-s − 0.316·10-s + 0.413·11-s + 0.380·13-s − 0.901·14-s + 0.250·16-s + 0.332·17-s − 0.314·19-s − 0.223·20-s + 0.292·22-s − 0.703·23-s + 0.200·25-s + 0.269·26-s − 0.637·28-s − 1.11·29-s + 0.492·31-s + 0.176·32-s + 0.235·34-s + 0.570·35-s − 0.164·37-s − 0.222·38-s − 0.158·40-s − 1.36·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 - 1.37T + 13T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 + 1.37T + 19T^{2} \) |
| 23 | \( 1 + 3.37T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 2.74T + 31T^{2} \) |
| 41 | \( 1 + 8.74T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 4.74T + 47T^{2} \) |
| 53 | \( 1 + 5.37T + 53T^{2} \) |
| 59 | \( 1 + 14.7T + 59T^{2} \) |
| 61 | \( 1 + 2.74T + 61T^{2} \) |
| 67 | \( 1 - 2.74T + 67T^{2} \) |
| 71 | \( 1 + 1.25T + 71T^{2} \) |
| 73 | \( 1 + 4.11T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 0.627T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.151112827566941020153314655912, −7.36716726103839439926041750527, −6.54859484058128516811337610090, −6.12314964877450324636714959788, −5.20782266792309859734974093443, −4.18185752622285749252271334572, −3.56666661532034164746079117739, −2.88675142955439155506145127817, −1.60753780957048935469755045442, 0,
1.60753780957048935469755045442, 2.88675142955439155506145127817, 3.56666661532034164746079117739, 4.18185752622285749252271334572, 5.20782266792309859734974093443, 6.12314964877450324636714959788, 6.54859484058128516811337610090, 7.36716726103839439926041750527, 8.151112827566941020153314655912
