| L(s) = 1 | − 2.45·2-s − 0.154·3-s + 4.01·4-s − 1.18·5-s + 0.377·6-s + 4.09·7-s − 4.94·8-s − 2.97·9-s + 2.89·10-s + 5.48·11-s − 0.618·12-s − 3.18·13-s − 10.0·14-s + 0.181·15-s + 4.09·16-s − 0.192·17-s + 7.30·18-s + 6.80·19-s − 4.74·20-s − 0.631·21-s − 13.4·22-s − 2.66·23-s + 0.762·24-s − 3.60·25-s + 7.82·26-s + 0.920·27-s + 16.4·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 0.0889·3-s + 2.00·4-s − 0.527·5-s + 0.154·6-s + 1.54·7-s − 1.74·8-s − 0.992·9-s + 0.915·10-s + 1.65·11-s − 0.178·12-s − 0.884·13-s − 2.68·14-s + 0.0469·15-s + 1.02·16-s − 0.0467·17-s + 1.72·18-s + 1.56·19-s − 1.05·20-s − 0.137·21-s − 2.86·22-s − 0.556·23-s + 0.155·24-s − 0.721·25-s + 1.53·26-s + 0.177·27-s + 3.11·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.45T + 2T^{2} \) |
| 3 | \( 1 + 0.154T + 3T^{2} \) |
| 5 | \( 1 + 1.18T + 5T^{2} \) |
| 7 | \( 1 - 4.09T + 7T^{2} \) |
| 11 | \( 1 - 5.48T + 11T^{2} \) |
| 13 | \( 1 + 3.18T + 13T^{2} \) |
| 17 | \( 1 + 0.192T + 17T^{2} \) |
| 19 | \( 1 - 6.80T + 19T^{2} \) |
| 23 | \( 1 + 2.66T + 23T^{2} \) |
| 29 | \( 1 + 9.24T + 29T^{2} \) |
| 31 | \( 1 - 4.63T + 31T^{2} \) |
| 37 | \( 1 + 5.89T + 37T^{2} \) |
| 41 | \( 1 - 4.56T + 41T^{2} \) |
| 43 | \( 1 + 4.29T + 43T^{2} \) |
| 47 | \( 1 + 5.19T + 47T^{2} \) |
| 53 | \( 1 + 7.66T + 53T^{2} \) |
| 59 | \( 1 + 7.73T + 59T^{2} \) |
| 61 | \( 1 - 1.67T + 61T^{2} \) |
| 67 | \( 1 + 1.81T + 67T^{2} \) |
| 71 | \( 1 + 6.15T + 71T^{2} \) |
| 73 | \( 1 + 4.56T + 73T^{2} \) |
| 79 | \( 1 - 9.09T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 7.31T + 89T^{2} \) |
| 97 | \( 1 - 2.45T + 97T^{2} \) |
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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
−8.093019511854612942221460392668, −7.65254315151030311136463021057, −7.05762145404326485993525604703, −6.08067258165062336422888243015, −5.21731501719487218158545417418, −4.23548552666126384943684475805, −3.14295368946900346072774237522, −1.92738649691115828668061781184, −1.30019337528302672482262445303, 0,
1.30019337528302672482262445303, 1.92738649691115828668061781184, 3.14295368946900346072774237522, 4.23548552666126384943684475805, 5.21731501719487218158545417418, 6.08067258165062336422888243015, 7.05762145404326485993525604703, 7.65254315151030311136463021057, 8.093019511854612942221460392668
