| L(s) = 1 | − 1.36·2-s − 0.363·3-s − 0.141·4-s − 3.14·5-s + 0.495·6-s + 2.91·8-s − 2.86·9-s + 4.28·10-s + 0.0513·12-s + 4.77·13-s + 1.14·15-s − 3.69·16-s + 4.77·17-s + 3.91·18-s + 7.00·19-s + 0.443·20-s + 5.14·23-s − 1.06·24-s + 4.86·25-s − 6.51·26-s + 2.13·27-s + 7.00·29-s − 1.55·30-s + 3.63·31-s − 0.797·32-s − 6.51·34-s + 0.405·36-s + ⋯ |
| L(s) = 1 | − 0.964·2-s − 0.209·3-s − 0.0706·4-s − 1.40·5-s + 0.202·6-s + 1.03·8-s − 0.955·9-s + 1.35·10-s + 0.0148·12-s + 1.32·13-s + 0.294·15-s − 0.924·16-s + 1.15·17-s + 0.921·18-s + 1.60·19-s + 0.0992·20-s + 1.07·23-s − 0.216·24-s + 0.973·25-s − 1.27·26-s + 0.410·27-s + 1.30·29-s − 0.284·30-s + 0.653·31-s − 0.141·32-s − 1.11·34-s + 0.0675·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7703697488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7703697488\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.36T + 2T^{2} \) |
| 3 | \( 1 + 0.363T + 3T^{2} \) |
| 5 | \( 1 + 3.14T + 5T^{2} \) |
| 13 | \( 1 - 4.77T + 13T^{2} \) |
| 17 | \( 1 - 4.77T + 17T^{2} \) |
| 19 | \( 1 - 7.00T + 19T^{2} \) |
| 23 | \( 1 - 5.14T + 23T^{2} \) |
| 29 | \( 1 - 7.00T + 29T^{2} \) |
| 31 | \( 1 - 3.63T + 31T^{2} \) |
| 37 | \( 1 + 9.86T + 37T^{2} \) |
| 41 | \( 1 - 3.22T + 41T^{2} \) |
| 43 | \( 1 - 4.28T + 43T^{2} \) |
| 47 | \( 1 - 0.778T + 47T^{2} \) |
| 53 | \( 1 - 2.28T + 53T^{2} \) |
| 59 | \( 1 - 0.363T + 59T^{2} \) |
| 61 | \( 1 + 3.22T + 61T^{2} \) |
| 67 | \( 1 + 6.59T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 3.22T + 73T^{2} \) |
| 79 | \( 1 + 3.71T + 79T^{2} \) |
| 83 | \( 1 + 1.55T + 83T^{2} \) |
| 89 | \( 1 - 5.58T + 89T^{2} \) |
| 97 | \( 1 + 6.15T + 97T^{2} \) |
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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
−8.177640581824599414631886228813, −7.59399629285060307517949035337, −7.03694123958476851385411642419, −5.97418502396502564632088931388, −5.19797355413528827064907491362, −4.44931249462997610935806203639, −3.47321808347612200726010995262, −3.03603504392090148691326895857, −1.23799549330227407322031050418, −0.65504734005695103753546160553,
0.65504734005695103753546160553, 1.23799549330227407322031050418, 3.03603504392090148691326895857, 3.47321808347612200726010995262, 4.44931249462997610935806203639, 5.19797355413528827064907491362, 5.97418502396502564632088931388, 7.03694123958476851385411642419, 7.59399629285060307517949035337, 8.177640581824599414631886228813
