L(s) = 1 | + 1.63·3-s + 2.87·5-s − 4.63·7-s − 0.335·9-s + 2.30·11-s − 5.47·13-s + 4.68·15-s + 3.36·17-s − 6.90·19-s − 7.57·21-s + 0.413·23-s + 3.23·25-s − 5.44·27-s + 3.71·29-s + 4.81·31-s + 3.75·33-s − 13.3·35-s − 9.04·37-s − 8.93·39-s − 1.24·41-s − 5.17·43-s − 0.963·45-s − 3.72·47-s + 14.5·49-s + 5.49·51-s − 4.50·53-s + 6.60·55-s + ⋯ |
L(s) = 1 | + 0.942·3-s + 1.28·5-s − 1.75·7-s − 0.111·9-s + 0.693·11-s − 1.51·13-s + 1.20·15-s + 0.816·17-s − 1.58·19-s − 1.65·21-s + 0.0861·23-s + 0.647·25-s − 1.04·27-s + 0.689·29-s + 0.864·31-s + 0.653·33-s − 2.25·35-s − 1.48·37-s − 1.43·39-s − 0.193·41-s − 0.789·43-s − 0.143·45-s − 0.543·47-s + 2.07·49-s + 0.769·51-s − 0.619·53-s + 0.890·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
good | 3 | \( 1 - 1.63T + 3T^{2} \) |
| 5 | \( 1 - 2.87T + 5T^{2} \) |
| 7 | \( 1 + 4.63T + 7T^{2} \) |
| 11 | \( 1 - 2.30T + 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 - 3.36T + 17T^{2} \) |
| 19 | \( 1 + 6.90T + 19T^{2} \) |
| 23 | \( 1 - 0.413T + 23T^{2} \) |
| 29 | \( 1 - 3.71T + 29T^{2} \) |
| 31 | \( 1 - 4.81T + 31T^{2} \) |
| 37 | \( 1 + 9.04T + 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 + 5.17T + 43T^{2} \) |
| 47 | \( 1 + 3.72T + 47T^{2} \) |
| 53 | \( 1 + 4.50T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 7.69T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 + 8.59T + 73T^{2} \) |
| 79 | \( 1 + 0.0752T + 79T^{2} \) |
| 83 | \( 1 + 9.54T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−8.321569430919381432821894336541, −7.18720431337177828929499341437, −6.58136067317627250664207454222, −6.04002884661726693482161824895, −5.18662656408914708883449556767, −4.07086840610606255789308052961, −3.09990104190750431966396989302, −2.63917676638376755326165028458, −1.75435273192840656288073561832, 0,
1.75435273192840656288073561832, 2.63917676638376755326165028458, 3.09990104190750431966396989302, 4.07086840610606255789308052961, 5.18662656408914708883449556767, 6.04002884661726693482161824895, 6.58136067317627250664207454222, 7.18720431337177828929499341437, 8.321569430919381432821894336541