| L(s) = 1 | − 1.29·2-s + 3-s − 0.332·4-s + 2.00·5-s − 1.29·6-s − 2.01·7-s + 3.01·8-s + 9-s − 2.58·10-s − 6.54·11-s − 0.332·12-s + 3.63·13-s + 2.59·14-s + 2.00·15-s − 3.22·16-s − 17-s − 1.29·18-s + 4.12·19-s − 0.666·20-s − 2.01·21-s + 8.44·22-s − 3.02·23-s + 3.01·24-s − 0.978·25-s − 4.69·26-s + 27-s + 0.668·28-s + ⋯ |
| L(s) = 1 | − 0.913·2-s + 0.577·3-s − 0.166·4-s + 0.896·5-s − 0.527·6-s − 0.760·7-s + 1.06·8-s + 0.333·9-s − 0.818·10-s − 1.97·11-s − 0.0959·12-s + 1.00·13-s + 0.694·14-s + 0.517·15-s − 0.806·16-s − 0.242·17-s − 0.304·18-s + 0.947·19-s − 0.149·20-s − 0.438·21-s + 1.80·22-s − 0.630·23-s + 0.614·24-s − 0.195·25-s − 0.919·26-s + 0.192·27-s + 0.126·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 1.29T + 2T^{2} \) |
| 5 | \( 1 - 2.00T + 5T^{2} \) |
| 7 | \( 1 + 2.01T + 7T^{2} \) |
| 11 | \( 1 + 6.54T + 11T^{2} \) |
| 13 | \( 1 - 3.63T + 13T^{2} \) |
| 19 | \( 1 - 4.12T + 19T^{2} \) |
| 23 | \( 1 + 3.02T + 23T^{2} \) |
| 29 | \( 1 - 4.97T + 29T^{2} \) |
| 31 | \( 1 + 5.55T + 31T^{2} \) |
| 37 | \( 1 - 6.55T + 37T^{2} \) |
| 41 | \( 1 - 0.296T + 41T^{2} \) |
| 43 | \( 1 + 4.09T + 43T^{2} \) |
| 47 | \( 1 - 1.28T + 47T^{2} \) |
| 53 | \( 1 - 5.43T + 53T^{2} \) |
| 59 | \( 1 + 3.34T + 59T^{2} \) |
| 61 | \( 1 + 5.90T + 61T^{2} \) |
| 67 | \( 1 + 4.55T + 67T^{2} \) |
| 71 | \( 1 + 3.22T + 71T^{2} \) |
| 73 | \( 1 - 5.13T + 73T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 0.258T + 89T^{2} \) |
| 97 | \( 1 - 0.411T + 97T^{2} \) |
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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
−8.092172759311725487138749241643, −7.72028774463324805127716266633, −6.78178672797224483843383780335, −5.84058427209486363212423755065, −5.22844155934248811868988189561, −4.20222102720378427280623759962, −3.15297129293284483346033720143, −2.37396939168173307827102992241, −1.35196824172583291852187303669, 0,
1.35196824172583291852187303669, 2.37396939168173307827102992241, 3.15297129293284483346033720143, 4.20222102720378427280623759962, 5.22844155934248811868988189561, 5.84058427209486363212423755065, 6.78178672797224483843383780335, 7.72028774463324805127716266633, 8.092172759311725487138749241643
