| L(s) = 1 | − 2.53·2-s − 3.24·3-s + 4.45·4-s + 3.59·5-s + 8.24·6-s + 4.74·7-s − 6.22·8-s + 7.52·9-s − 9.12·10-s + 4.76·11-s − 14.4·12-s + 3.36·13-s − 12.0·14-s − 11.6·15-s + 6.90·16-s + 5.35·17-s − 19.1·18-s − 2.01·19-s + 15.9·20-s − 15.3·21-s − 12.0·22-s − 2.21·23-s + 20.1·24-s + 7.90·25-s − 8.55·26-s − 14.6·27-s + 21.1·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s − 1.87·3-s + 2.22·4-s + 1.60·5-s + 3.36·6-s + 1.79·7-s − 2.20·8-s + 2.50·9-s − 2.88·10-s + 1.43·11-s − 4.16·12-s + 0.933·13-s − 3.21·14-s − 3.00·15-s + 1.72·16-s + 1.29·17-s − 4.50·18-s − 0.462·19-s + 3.57·20-s − 3.35·21-s − 2.57·22-s − 0.461·23-s + 4.12·24-s + 1.58·25-s − 1.67·26-s − 2.82·27-s + 3.98·28-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
| \(L(1)\) |
\(\approx\) |
\(0.9951583918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9951583918\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.53T + 2T^{2} \) |
| 3 | \( 1 + 3.24T + 3T^{2} \) |
| 5 | \( 1 - 3.59T + 5T^{2} \) |
| 7 | \( 1 - 4.74T + 7T^{2} \) |
| 11 | \( 1 - 4.76T + 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 - 5.35T + 17T^{2} \) |
| 19 | \( 1 + 2.01T + 19T^{2} \) |
| 23 | \( 1 + 2.21T + 23T^{2} \) |
| 29 | \( 1 + 3.92T + 29T^{2} \) |
| 31 | \( 1 + 6.33T + 31T^{2} \) |
| 37 | \( 1 - 9.40T + 37T^{2} \) |
| 41 | \( 1 - 0.410T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 2.33T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 6.43T + 59T^{2} \) |
| 61 | \( 1 + 9.91T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 3.56T + 73T^{2} \) |
| 79 | \( 1 + 1.58T + 79T^{2} \) |
| 83 | \( 1 - 1.22T + 83T^{2} \) |
| 89 | \( 1 + 3.37T + 89T^{2} \) |
| 97 | \( 1 - 2.68T + 97T^{2} \) |
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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
−8.538641946094266069463921322852, −7.76135493007652101837564122920, −6.93159084110781326980756035846, −6.28167948645956616140751072022, −5.76682166342988823272796817079, −5.16182802121324376388950875591, −4.02430959171958756295839076062, −1.94264004714298937462499559000, −1.48372445300105090601720538721, −0.985079040653652791390862313131,
0.985079040653652791390862313131, 1.48372445300105090601720538721, 1.94264004714298937462499559000, 4.02430959171958756295839076062, 5.16182802121324376388950875591, 5.76682166342988823272796817079, 6.28167948645956616140751072022, 6.93159084110781326980756035846, 7.76135493007652101837564122920, 8.538641946094266069463921322852
