| L(s) = 1 | − 2.70·2-s − 1.68·3-s + 5.32·4-s + 1.67·5-s + 4.55·6-s + 1.84·7-s − 8.98·8-s − 0.166·9-s − 4.53·10-s + 0.558·11-s − 8.95·12-s − 5.94·13-s − 4.98·14-s − 2.81·15-s + 13.6·16-s + 17-s + 0.451·18-s + 3.04·19-s + 8.91·20-s − 3.10·21-s − 1.51·22-s − 5.10·23-s + 15.1·24-s − 2.19·25-s + 16.0·26-s + 5.33·27-s + 9.79·28-s + ⋯ |
| L(s) = 1 | − 1.91·2-s − 0.971·3-s + 2.66·4-s + 0.748·5-s + 1.85·6-s + 0.696·7-s − 3.17·8-s − 0.0556·9-s − 1.43·10-s + 0.168·11-s − 2.58·12-s − 1.64·13-s − 1.33·14-s − 0.727·15-s + 3.41·16-s + 0.242·17-s + 0.106·18-s + 0.699·19-s + 1.99·20-s − 0.676·21-s − 0.322·22-s − 1.06·23-s + 3.08·24-s − 0.439·25-s + 3.15·26-s + 1.02·27-s + 1.85·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4215010740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4215010740\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 353 | \( 1 + T \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 3 | \( 1 + 1.68T + 3T^{2} \) |
| 5 | \( 1 - 1.67T + 5T^{2} \) |
| 7 | \( 1 - 1.84T + 7T^{2} \) |
| 11 | \( 1 - 0.558T + 11T^{2} \) |
| 13 | \( 1 + 5.94T + 13T^{2} \) |
| 19 | \( 1 - 3.04T + 19T^{2} \) |
| 23 | \( 1 + 5.10T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 7.50T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 + 9.13T + 41T^{2} \) |
| 43 | \( 1 + 6.30T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 5.57T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 - 9.95T + 61T^{2} \) |
| 67 | \( 1 - 2.73T + 67T^{2} \) |
| 71 | \( 1 + 4.00T + 71T^{2} \) |
| 73 | \( 1 + 8.56T + 73T^{2} \) |
| 79 | \( 1 - 4.26T + 79T^{2} \) |
| 83 | \( 1 - 2.46T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 17.2T + 97T^{2} \) |
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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
−8.165500860672422259213410424963, −7.51360105764856098024740292583, −6.76212795330740089033547708185, −6.25292590853781262324833041918, −5.39441949751610796396559675711, −4.88658999842604532751762558945, −3.20273683523144017385520692199, −2.22214577474503317780820077689, −1.60665218556970549731229891304, −0.47769849447607621965972525545,
0.47769849447607621965972525545, 1.60665218556970549731229891304, 2.22214577474503317780820077689, 3.20273683523144017385520692199, 4.88658999842604532751762558945, 5.39441949751610796396559675711, 6.25292590853781262324833041918, 6.76212795330740089033547708185, 7.51360105764856098024740292583, 8.165500860672422259213410424963
