| L(s) = 1 | + 2-s + 1.73·3-s + 4-s − 3.78·5-s + 1.73·6-s − 2.72·7-s + 8-s − 0.000352·9-s − 3.78·10-s + 0.672·11-s + 1.73·12-s + 0.231·13-s − 2.72·14-s − 6.55·15-s + 16-s + 3.23·17-s − 0.000352·18-s − 7.61·19-s − 3.78·20-s − 4.71·21-s + 0.672·22-s − 0.749·23-s + 1.73·24-s + 9.30·25-s + 0.231·26-s − 5.19·27-s − 2.72·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.999·3-s + 0.5·4-s − 1.69·5-s + 0.707·6-s − 1.02·7-s + 0.353·8-s − 0.000117·9-s − 1.19·10-s + 0.202·11-s + 0.499·12-s + 0.0641·13-s − 0.727·14-s − 1.69·15-s + 0.250·16-s + 0.783·17-s − 8.31e − 5·18-s − 1.74·19-s − 0.845·20-s − 1.02·21-s + 0.143·22-s − 0.156·23-s + 0.353·24-s + 1.86·25-s + 0.0453·26-s − 1.00·27-s − 0.514·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.384280193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.384280193\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 + 3.78T + 5T^{2} \) |
| 7 | \( 1 + 2.72T + 7T^{2} \) |
| 11 | \( 1 - 0.672T + 11T^{2} \) |
| 13 | \( 1 - 0.231T + 13T^{2} \) |
| 17 | \( 1 - 3.23T + 17T^{2} \) |
| 19 | \( 1 + 7.61T + 19T^{2} \) |
| 23 | \( 1 + 0.749T + 23T^{2} \) |
| 29 | \( 1 - 5.96T + 29T^{2} \) |
| 37 | \( 1 - 8.58T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 7.17T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 + 6.37T + 53T^{2} \) |
| 59 | \( 1 - 4.55T + 59T^{2} \) |
| 61 | \( 1 + 3.19T + 61T^{2} \) |
| 67 | \( 1 - 7.44T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 - 7.85T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 4.16T + 83T^{2} \) |
| 89 | \( 1 + 6.83T + 89T^{2} \) |
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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
−8.014914842792300669329073247961, −7.54573644412385833256911065097, −6.57354654139379412076240794218, −6.17882556119708126936273430573, −4.93763569629240354723063958488, −4.05934780205606355714003253276, −3.73970770706445157560039925012, −2.99088777479053764263623358786, −2.35451846375401867285434137789, −0.66395479868707655185318052723,
0.66395479868707655185318052723, 2.35451846375401867285434137789, 2.99088777479053764263623358786, 3.73970770706445157560039925012, 4.05934780205606355714003253276, 4.93763569629240354723063958488, 6.17882556119708126936273430573, 6.57354654139379412076240794218, 7.54573644412385833256911065097, 8.014914842792300669329073247961
