| L(s) = 1 | − 0.886·2-s − 3.35·3-s − 1.21·4-s − 3.79·5-s + 2.97·6-s − 4.56·7-s + 2.84·8-s + 8.26·9-s + 3.36·10-s + 3.91·11-s + 4.07·12-s − 4.65·13-s + 4.04·14-s + 12.7·15-s − 0.0975·16-s + 17-s − 7.32·18-s − 0.168·19-s + 4.61·20-s + 15.3·21-s − 3.46·22-s + 2.29·23-s − 9.56·24-s + 9.42·25-s + 4.12·26-s − 17.6·27-s + 5.53·28-s + ⋯ |
| L(s) = 1 | − 0.626·2-s − 1.93·3-s − 0.607·4-s − 1.69·5-s + 1.21·6-s − 1.72·7-s + 1.00·8-s + 2.75·9-s + 1.06·10-s + 1.17·11-s + 1.17·12-s − 1.29·13-s + 1.08·14-s + 3.29·15-s − 0.0243·16-s + 0.242·17-s − 1.72·18-s − 0.0386·19-s + 1.03·20-s + 3.33·21-s − 0.739·22-s + 0.478·23-s − 1.95·24-s + 1.88·25-s + 0.808·26-s − 3.39·27-s + 1.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 + 0.886T + 2T^{2} \) |
| 3 | \( 1 + 3.35T + 3T^{2} \) |
| 5 | \( 1 + 3.79T + 5T^{2} \) |
| 7 | \( 1 + 4.56T + 7T^{2} \) |
| 11 | \( 1 - 3.91T + 11T^{2} \) |
| 13 | \( 1 + 4.65T + 13T^{2} \) |
| 19 | \( 1 + 0.168T + 19T^{2} \) |
| 23 | \( 1 - 2.29T + 23T^{2} \) |
| 29 | \( 1 - 1.03T + 29T^{2} \) |
| 31 | \( 1 - 1.22T + 31T^{2} \) |
| 37 | \( 1 + 5.79T + 37T^{2} \) |
| 41 | \( 1 + 2.92T + 41T^{2} \) |
| 43 | \( 1 - 8.31T + 43T^{2} \) |
| 47 | \( 1 - 6.36T + 47T^{2} \) |
| 53 | \( 1 + 9.27T + 53T^{2} \) |
| 61 | \( 1 + 4.07T + 61T^{2} \) |
| 67 | \( 1 - 8.71T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 1.39T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 4.71T + 83T^{2} \) |
| 89 | \( 1 + 4.92T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−9.716251255818472618407651108567, −8.966611999535779394465055392520, −7.55737133299301944878412072110, −7.06124618799932755303339097501, −6.33678897473865506820971363423, −5.11186169705653967787848694720, −4.29763240810814151572853015251, −3.58829404354048512652502267686, −0.814946723971357906412223086233, 0,
0.814946723971357906412223086233, 3.58829404354048512652502267686, 4.29763240810814151572853015251, 5.11186169705653967787848694720, 6.33678897473865506820971363423, 7.06124618799932755303339097501, 7.55737133299301944878412072110, 8.966611999535779394465055392520, 9.716251255818472618407651108567
