| L(s) = 1 | + 1.68·2-s − 1.06·3-s + 0.844·4-s − 5-s − 1.79·6-s + 7-s − 1.94·8-s − 1.86·9-s − 1.68·10-s + 0.272·11-s − 0.898·12-s − 0.305·13-s + 1.68·14-s + 1.06·15-s − 4.97·16-s + 6.91·17-s − 3.15·18-s − 1.70·19-s − 0.844·20-s − 1.06·21-s + 0.459·22-s + 2.62·23-s + 2.07·24-s + 25-s − 0.515·26-s + 5.17·27-s + 0.844·28-s + ⋯ |
| L(s) = 1 | + 1.19·2-s − 0.614·3-s + 0.422·4-s − 0.447·5-s − 0.732·6-s + 0.377·7-s − 0.688·8-s − 0.622·9-s − 0.533·10-s + 0.0821·11-s − 0.259·12-s − 0.0848·13-s + 0.450·14-s + 0.274·15-s − 1.24·16-s + 1.67·17-s − 0.742·18-s − 0.392·19-s − 0.188·20-s − 0.232·21-s + 0.0979·22-s + 0.547·23-s + 0.423·24-s + 0.200·25-s − 0.101·26-s + 0.996·27-s + 0.159·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 1.68T + 2T^{2} \) |
| 3 | \( 1 + 1.06T + 3T^{2} \) |
| 11 | \( 1 - 0.272T + 11T^{2} \) |
| 13 | \( 1 + 0.305T + 13T^{2} \) |
| 17 | \( 1 - 6.91T + 17T^{2} \) |
| 19 | \( 1 + 1.70T + 19T^{2} \) |
| 23 | \( 1 - 2.62T + 23T^{2} \) |
| 29 | \( 1 - 1.14T + 29T^{2} \) |
| 31 | \( 1 + 3.35T + 31T^{2} \) |
| 37 | \( 1 - 9.50T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 4.67T + 43T^{2} \) |
| 47 | \( 1 + 1.30T + 47T^{2} \) |
| 53 | \( 1 + 7.61T + 53T^{2} \) |
| 59 | \( 1 - 5.13T + 59T^{2} \) |
| 61 | \( 1 + 8.03T + 61T^{2} \) |
| 67 | \( 1 + 8.12T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 6.13T + 79T^{2} \) |
| 83 | \( 1 - 4.02T + 83T^{2} \) |
| 89 | \( 1 + 9.83T + 89T^{2} \) |
| 97 | \( 1 + 3.70T + 97T^{2} \) |
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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
−7.38592193404653804601221427976, −6.45376793381411417348867738527, −5.94554280894847151086115544094, −5.24641532155847143082717833173, −4.83815152131537776917355823691, −3.99771417819947060328123279198, −3.26978668602081047353053567560, −2.63148891156214220878051948995, −1.22167276923703166446471774900, 0,
1.22167276923703166446471774900, 2.63148891156214220878051948995, 3.26978668602081047353053567560, 3.99771417819947060328123279198, 4.83815152131537776917355823691, 5.24641532155847143082717833173, 5.94554280894847151086115544094, 6.45376793381411417348867738527, 7.38592193404653804601221427976
