L(s) = 1 | − 2-s − 2.49·3-s + 4-s − 3.77·5-s + 2.49·6-s − 0.508·7-s − 8-s + 3.24·9-s + 3.77·10-s + 4.72·11-s − 2.49·12-s − 1.56·13-s + 0.508·14-s + 9.44·15-s + 16-s − 1.70·17-s − 3.24·18-s − 7.33·19-s − 3.77·20-s + 1.27·21-s − 4.72·22-s + 6.82·23-s + 2.49·24-s + 9.27·25-s + 1.56·26-s − 0.613·27-s − 0.508·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.44·3-s + 0.5·4-s − 1.68·5-s + 1.02·6-s − 0.192·7-s − 0.353·8-s + 1.08·9-s + 1.19·10-s + 1.42·11-s − 0.721·12-s − 0.434·13-s + 0.135·14-s + 2.43·15-s + 0.250·16-s − 0.414·17-s − 0.764·18-s − 1.68·19-s − 0.844·20-s + 0.277·21-s − 1.00·22-s + 1.42·23-s + 0.510·24-s + 1.85·25-s + 0.307·26-s − 0.118·27-s − 0.0961·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4040202603\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4040202603\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 4021 | \( 1+O(T) \) |
good | 3 | \( 1 + 2.49T + 3T^{2} \) |
| 5 | \( 1 + 3.77T + 5T^{2} \) |
| 7 | \( 1 + 0.508T + 7T^{2} \) |
| 11 | \( 1 - 4.72T + 11T^{2} \) |
| 13 | \( 1 + 1.56T + 13T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 + 7.33T + 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 - 6.46T + 29T^{2} \) |
| 31 | \( 1 + 0.0712T + 31T^{2} \) |
| 37 | \( 1 - 9.42T + 37T^{2} \) |
| 41 | \( 1 + 5.14T + 41T^{2} \) |
| 43 | \( 1 - 1.98T + 43T^{2} \) |
| 47 | \( 1 - 5.36T + 47T^{2} \) |
| 53 | \( 1 - 0.630T + 53T^{2} \) |
| 59 | \( 1 + 3.96T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 0.192T + 67T^{2} \) |
| 71 | \( 1 - 4.53T + 71T^{2} \) |
| 73 | \( 1 - 0.588T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 - 6.65T + 83T^{2} \) |
| 89 | \( 1 + 0.667T + 89T^{2} \) |
| 97 | \( 1 - 2.10T + 97T^{2} \) |
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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
−7.77069577524154547120085118883, −6.90113261808864433573150192060, −6.69457451418856899990700104762, −6.05060146811685340604469113634, −4.85815380224784965865638329071, −4.41579454216184386033100509450, −3.69584755555899648740268942276, −2.63069203549623263948722084891, −1.18167633176091716381901507190, −0.44722982744167185516629886960,
0.44722982744167185516629886960, 1.18167633176091716381901507190, 2.63069203549623263948722084891, 3.69584755555899648740268942276, 4.41579454216184386033100509450, 4.85815380224784965865638329071, 6.05060146811685340604469113634, 6.69457451418856899990700104762, 6.90113261808864433573150192060, 7.77069577524154547120085118883