| L(s) = 1 | − 2.54·2-s − 3.05·3-s + 4.49·4-s + 1.69·5-s + 7.78·6-s − 1.95·7-s − 6.34·8-s + 6.34·9-s − 4.32·10-s + 1.27·11-s − 13.7·12-s + 2.11·13-s + 4.97·14-s − 5.18·15-s + 7.19·16-s − 4.74·17-s − 16.1·18-s + 6.91·19-s + 7.62·20-s + 5.96·21-s − 3.25·22-s − 0.312·23-s + 19.4·24-s − 2.11·25-s − 5.38·26-s − 10.2·27-s − 8.76·28-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 1.76·3-s + 2.24·4-s + 0.759·5-s + 3.17·6-s − 0.737·7-s − 2.24·8-s + 2.11·9-s − 1.36·10-s + 0.385·11-s − 3.96·12-s + 0.585·13-s + 1.32·14-s − 1.33·15-s + 1.79·16-s − 1.15·17-s − 3.80·18-s + 1.58·19-s + 1.70·20-s + 1.30·21-s − 0.694·22-s − 0.0652·23-s + 3.96·24-s − 0.423·25-s − 1.05·26-s − 1.96·27-s − 1.65·28-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 | \( 1 + 2.54T + 2T^{2} \) |
| 3 | \( 1 + 3.05T + 3T^{2} \) |
| 5 | \( 1 - 1.69T + 5T^{2} \) |
| 7 | \( 1 + 1.95T + 7T^{2} \) |
| 11 | \( 1 - 1.27T + 11T^{2} \) |
| 13 | \( 1 - 2.11T + 13T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 - 6.91T + 19T^{2} \) |
| 23 | \( 1 + 0.312T + 23T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 31 | \( 1 - 6.09T + 31T^{2} \) |
| 37 | \( 1 + 5.58T + 37T^{2} \) |
| 41 | \( 1 + 8.81T + 41T^{2} \) |
| 43 | \( 1 - 2.76T + 43T^{2} \) |
| 47 | \( 1 - 1.05T + 47T^{2} \) |
| 53 | \( 1 + 0.198T + 53T^{2} \) |
| 59 | \( 1 + 1.29T + 59T^{2} \) |
| 61 | \( 1 - 1.28T + 61T^{2} \) |
| 67 | \( 1 - 4.45T + 67T^{2} \) |
| 71 | \( 1 + 3.26T + 71T^{2} \) |
| 73 | \( 1 + 3.95T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 5.43T + 89T^{2} \) |
| 97 | \( 1 + 8.36T + 97T^{2} \) |
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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
−8.214426151217950511949405240083, −6.98709594459201047297583040885, −6.86578045693176984015634199948, −6.07060699314708615539319647632, −5.59256345656242133606564480817, −4.47842458587296409277084943215, −3.09344818161305169186985975790, −1.79478338751228312769182725444, −1.03528122521271794592734562251, 0,
1.03528122521271794592734562251, 1.79478338751228312769182725444, 3.09344818161305169186985975790, 4.47842458587296409277084943215, 5.59256345656242133606564480817, 6.07060699314708615539319647632, 6.86578045693176984015634199948, 6.98709594459201047297583040885, 8.214426151217950511949405240083
