| L(s) = 1 | − 2.64·2-s + 1.02·3-s + 4.98·4-s + 2.32·5-s − 2.70·6-s − 0.493·7-s − 7.87·8-s − 1.94·9-s − 6.14·10-s + 2.96·11-s + 5.10·12-s + 0.776·13-s + 1.30·14-s + 2.38·15-s + 10.8·16-s + 6.61·17-s + 5.15·18-s − 7.31·19-s + 11.5·20-s − 0.506·21-s − 7.82·22-s − 8.89·23-s − 8.07·24-s + 0.413·25-s − 2.05·26-s − 5.07·27-s − 2.45·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 0.591·3-s + 2.49·4-s + 1.04·5-s − 1.10·6-s − 0.186·7-s − 2.78·8-s − 0.649·9-s − 1.94·10-s + 0.892·11-s + 1.47·12-s + 0.215·13-s + 0.348·14-s + 0.615·15-s + 2.71·16-s + 1.60·17-s + 1.21·18-s − 1.67·19-s + 2.59·20-s − 0.110·21-s − 1.66·22-s − 1.85·23-s − 1.64·24-s + 0.0827·25-s − 0.402·26-s − 0.976·27-s − 0.464·28-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 | \( 1 + 2.64T + 2T^{2} \) |
| 3 | \( 1 - 1.02T + 3T^{2} \) |
| 5 | \( 1 - 2.32T + 5T^{2} \) |
| 7 | \( 1 + 0.493T + 7T^{2} \) |
| 11 | \( 1 - 2.96T + 11T^{2} \) |
| 13 | \( 1 - 0.776T + 13T^{2} \) |
| 17 | \( 1 - 6.61T + 17T^{2} \) |
| 19 | \( 1 + 7.31T + 19T^{2} \) |
| 23 | \( 1 + 8.89T + 23T^{2} \) |
| 29 | \( 1 + 2.08T + 29T^{2} \) |
| 31 | \( 1 - 0.878T + 31T^{2} \) |
| 37 | \( 1 - 4.25T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 7.88T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 2.76T + 59T^{2} \) |
| 61 | \( 1 - 8.06T + 61T^{2} \) |
| 67 | \( 1 + 0.138T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 5.70T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 6.74T + 97T^{2} \) |
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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
−8.243062594797804353014310365460, −7.78542766577878475213799765016, −6.68474256781137785732640535766, −6.17234214197292429828499677737, −5.61241040510406088078197536193, −3.90084756060065004858720434352, −2.96368583564874653619718583361, −2.02869019069368612594743077718, −1.51749838412792102819459321654, 0,
1.51749838412792102819459321654, 2.02869019069368612594743077718, 2.96368583564874653619718583361, 3.90084756060065004858720434352, 5.61241040510406088078197536193, 6.17234214197292429828499677737, 6.68474256781137785732640535766, 7.78542766577878475213799765016, 8.243062594797804353014310365460
