| L(s) = 1 | − 2-s − 2.42·3-s + 4-s − 1.19·5-s + 2.42·6-s − 3.77·7-s − 8-s + 2.86·9-s + 1.19·10-s − 4.05·11-s − 2.42·12-s + 4.55·13-s + 3.77·14-s + 2.88·15-s + 16-s − 1.21·17-s − 2.86·18-s + 3.72·19-s − 1.19·20-s + 9.13·21-s + 4.05·22-s + 5.44·23-s + 2.42·24-s − 3.58·25-s − 4.55·26-s + 0.334·27-s − 3.77·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.39·3-s + 0.5·4-s − 0.532·5-s + 0.988·6-s − 1.42·7-s − 0.353·8-s + 0.953·9-s + 0.376·10-s − 1.22·11-s − 0.698·12-s + 1.26·13-s + 1.00·14-s + 0.744·15-s + 0.250·16-s − 0.294·17-s − 0.674·18-s + 0.853·19-s − 0.266·20-s + 1.99·21-s + 0.864·22-s + 1.13·23-s + 0.494·24-s − 0.716·25-s − 0.893·26-s + 0.0643·27-s − 0.713·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2337024935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2337024935\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 4001 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.42T + 3T^{2} \) |
| 5 | \( 1 + 1.19T + 5T^{2} \) |
| 7 | \( 1 + 3.77T + 7T^{2} \) |
| 11 | \( 1 + 4.05T + 11T^{2} \) |
| 13 | \( 1 - 4.55T + 13T^{2} \) |
| 17 | \( 1 + 1.21T + 17T^{2} \) |
| 19 | \( 1 - 3.72T + 19T^{2} \) |
| 23 | \( 1 - 5.44T + 23T^{2} \) |
| 29 | \( 1 + 4.71T + 29T^{2} \) |
| 31 | \( 1 - 4.49T + 31T^{2} \) |
| 37 | \( 1 + 3.22T + 37T^{2} \) |
| 41 | \( 1 - 2.91T + 41T^{2} \) |
| 43 | \( 1 + 0.507T + 43T^{2} \) |
| 47 | \( 1 - 1.10T + 47T^{2} \) |
| 53 | \( 1 - 0.544T + 53T^{2} \) |
| 59 | \( 1 + 1.10T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 + 3.02T + 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 + 6.91T + 73T^{2} \) |
| 79 | \( 1 + 9.08T + 79T^{2} \) |
| 83 | \( 1 + 2.39T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 + 6.06T + 97T^{2} \) |
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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
−7.72030363255664713557583230770, −7.06097060171054525121681049305, −6.48518272914192602542763866910, −5.81046329088461270710397821391, −5.38462294598267141111012240650, −4.33882619359074580379930915261, −3.38148328273005691295167143546, −2.77077013767614887287906621959, −1.28885030985261477023721458972, −0.31377546227763260983840831189,
0.31377546227763260983840831189, 1.28885030985261477023721458972, 2.77077013767614887287906621959, 3.38148328273005691295167143546, 4.33882619359074580379930915261, 5.38462294598267141111012240650, 5.81046329088461270710397821391, 6.48518272914192602542763866910, 7.06097060171054525121681049305, 7.72030363255664713557583230770
