| L(s) = 1 | − 2.61·2-s − 2.40·3-s + 4.84·4-s + 1.76·5-s + 6.30·6-s − 1.48·7-s − 7.45·8-s + 2.79·9-s − 4.60·10-s − 4.43·11-s − 11.6·12-s + 3.63·13-s + 3.89·14-s − 4.24·15-s + 9.80·16-s − 3.52·17-s − 7.31·18-s − 7.25·19-s + 8.53·20-s + 3.58·21-s + 11.5·22-s − 0.195·23-s + 17.9·24-s − 1.89·25-s − 9.51·26-s + 0.489·27-s − 7.21·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 1.39·3-s + 2.42·4-s + 0.787·5-s + 2.57·6-s − 0.562·7-s − 2.63·8-s + 0.932·9-s − 1.45·10-s − 1.33·11-s − 3.36·12-s + 1.00·13-s + 1.04·14-s − 1.09·15-s + 2.45·16-s − 0.853·17-s − 1.72·18-s − 1.66·19-s + 1.90·20-s + 0.781·21-s + 2.47·22-s − 0.0407·23-s + 3.66·24-s − 0.379·25-s − 1.86·26-s + 0.0941·27-s − 1.36·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07673809736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07673809736\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4003 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 + 2.40T + 3T^{2} \) |
| 5 | \( 1 - 1.76T + 5T^{2} \) |
| 7 | \( 1 + 1.48T + 7T^{2} \) |
| 11 | \( 1 + 4.43T + 11T^{2} \) |
| 13 | \( 1 - 3.63T + 13T^{2} \) |
| 17 | \( 1 + 3.52T + 17T^{2} \) |
| 19 | \( 1 + 7.25T + 19T^{2} \) |
| 23 | \( 1 + 0.195T + 23T^{2} \) |
| 29 | \( 1 + 5.52T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 3.71T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 0.699T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 0.122T + 59T^{2} \) |
| 61 | \( 1 - 6.99T + 61T^{2} \) |
| 67 | \( 1 + 3.16T + 67T^{2} \) |
| 71 | \( 1 + 6.44T + 71T^{2} \) |
| 73 | \( 1 - 6.88T + 73T^{2} \) |
| 79 | \( 1 + 1.12T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 9.84T + 97T^{2} \) |
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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
−8.690910358819341238096835511339, −7.79162563225635484815595299250, −6.88691962655952897381218381993, −6.43775868029142186344349179349, −5.80221308721426568547580996757, −5.17518135326565953718640793583, −3.66904841190371751989657799911, −2.30075033151680181582881652430, −1.70108897290349034020286365686, −0.21549369489152301992065364821,
0.21549369489152301992065364821, 1.70108897290349034020286365686, 2.30075033151680181582881652430, 3.66904841190371751989657799911, 5.17518135326565953718640793583, 5.80221308721426568547580996757, 6.43775868029142186344349179349, 6.88691962655952897381218381993, 7.79162563225635484815595299250, 8.690910358819341238096835511339
