| L(s) = 1 | − 1.47·2-s − 3-s + 0.182·4-s − 3.64·5-s + 1.47·6-s − 4.01·7-s + 2.68·8-s + 9-s + 5.38·10-s + 1.40·11-s − 0.182·12-s − 13-s + 5.93·14-s + 3.64·15-s − 4.33·16-s − 3.90·17-s − 1.47·18-s − 5.03·19-s − 0.665·20-s + 4.01·21-s − 2.08·22-s − 2.20·23-s − 2.68·24-s + 8.28·25-s + 1.47·26-s − 27-s − 0.733·28-s + ⋯ |
| L(s) = 1 | − 1.04·2-s − 0.577·3-s + 0.0913·4-s − 1.63·5-s + 0.603·6-s − 1.51·7-s + 0.949·8-s + 0.333·9-s + 1.70·10-s + 0.424·11-s − 0.0527·12-s − 0.277·13-s + 1.58·14-s + 0.941·15-s − 1.08·16-s − 0.945·17-s − 0.348·18-s − 1.15·19-s − 0.148·20-s + 0.876·21-s − 0.443·22-s − 0.458·23-s − 0.548·24-s + 1.65·25-s + 0.289·26-s − 0.192·27-s − 0.138·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 1.47T + 2T^{2} \) |
| 5 | \( 1 + 3.64T + 5T^{2} \) |
| 7 | \( 1 + 4.01T + 7T^{2} \) |
| 11 | \( 1 - 1.40T + 11T^{2} \) |
| 17 | \( 1 + 3.90T + 17T^{2} \) |
| 19 | \( 1 + 5.03T + 19T^{2} \) |
| 23 | \( 1 + 2.20T + 23T^{2} \) |
| 29 | \( 1 + 0.586T + 29T^{2} \) |
| 31 | \( 1 + 0.278T + 31T^{2} \) |
| 37 | \( 1 - 3.21T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 - 1.60T + 43T^{2} \) |
| 47 | \( 1 + 5.58T + 47T^{2} \) |
| 53 | \( 1 + 7.05T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 - 8.01T + 61T^{2} \) |
| 67 | \( 1 - 8.85T + 67T^{2} \) |
| 71 | \( 1 + 1.06T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 3.44T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 18.3T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−8.008442581524638223540171341345, −7.55824280904522858270275075884, −6.63442472132707118775163454697, −6.33598475092330241992733631153, −4.87952732406591558022012129912, −4.14332037201984570764702319926, −3.64511346125349771798696958215, −2.33113820409013607635088240343, −0.70026439060589582973756993061, 0,
0.70026439060589582973756993061, 2.33113820409013607635088240343, 3.64511346125349771798696958215, 4.14332037201984570764702319926, 4.87952732406591558022012129912, 6.33598475092330241992733631153, 6.63442472132707118775163454697, 7.55824280904522858270275075884, 8.008442581524638223540171341345
