| L(s) = 1 | + 2-s + 1.91·3-s + 4-s − 1.19·5-s + 1.91·6-s − 2.67·7-s + 8-s + 0.657·9-s − 1.19·10-s + 4.98·11-s + 1.91·12-s − 3.88·13-s − 2.67·14-s − 2.28·15-s + 16-s − 5.50·17-s + 0.657·18-s − 7.93·19-s − 1.19·20-s − 5.11·21-s + 4.98·22-s + 4.47·23-s + 1.91·24-s − 3.57·25-s − 3.88·26-s − 4.47·27-s − 2.67·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.10·3-s + 0.5·4-s − 0.533·5-s + 0.780·6-s − 1.01·7-s + 0.353·8-s + 0.219·9-s − 0.377·10-s + 1.50·11-s + 0.552·12-s − 1.07·13-s − 0.715·14-s − 0.588·15-s + 0.250·16-s − 1.33·17-s + 0.154·18-s − 1.82·19-s − 0.266·20-s − 1.11·21-s + 1.06·22-s + 0.934·23-s + 0.390·24-s − 0.715·25-s − 0.762·26-s − 0.862·27-s − 0.505·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 - 1.91T + 3T^{2} \) |
| 5 | \( 1 + 1.19T + 5T^{2} \) |
| 7 | \( 1 + 2.67T + 7T^{2} \) |
| 11 | \( 1 - 4.98T + 11T^{2} \) |
| 13 | \( 1 + 3.88T + 13T^{2} \) |
| 17 | \( 1 + 5.50T + 17T^{2} \) |
| 19 | \( 1 + 7.93T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 1.48T + 29T^{2} \) |
| 31 | \( 1 + 0.00388T + 31T^{2} \) |
| 37 | \( 1 + 2.18T + 37T^{2} \) |
| 41 | \( 1 + 3.84T + 41T^{2} \) |
| 43 | \( 1 - 8.09T + 43T^{2} \) |
| 47 | \( 1 + 6.80T + 47T^{2} \) |
| 53 | \( 1 - 1.44T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 9.77T + 61T^{2} \) |
| 67 | \( 1 - 0.127T + 67T^{2} \) |
| 71 | \( 1 + 1.98T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 7.12T + 83T^{2} \) |
| 89 | \( 1 + 8.36T + 89T^{2} \) |
| 97 | \( 1 - 6.90T + 97T^{2} \) |
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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
−8.114634789200612863253566757643, −7.16631787196790214753818444015, −6.67119700272372838904842562463, −6.04843769305424485644606747257, −4.74615098115626916980099117263, −4.05302676098527502881083525006, −3.52595945365865157320168207349, −2.62542154732538400973352373217, −1.92030925294832951793204232231, 0,
1.92030925294832951793204232231, 2.62542154732538400973352373217, 3.52595945365865157320168207349, 4.05302676098527502881083525006, 4.74615098115626916980099117263, 6.04843769305424485644606747257, 6.67119700272372838904842562463, 7.16631787196790214753818444015, 8.114634789200612863253566757643
