L(s) = 1 | + 0.773·2-s − 2.92·3-s − 1.40·4-s − 5-s − 2.26·6-s + 4.94·7-s − 2.63·8-s + 5.55·9-s − 0.773·10-s − 11-s + 4.09·12-s − 4.04·13-s + 3.82·14-s + 2.92·15-s + 0.767·16-s − 5.75·17-s + 4.29·18-s + 5.65·19-s + 1.40·20-s − 14.4·21-s − 0.773·22-s − 6.35·23-s + 7.69·24-s + 25-s − 3.12·26-s − 7.47·27-s − 6.92·28-s + ⋯ |
L(s) = 1 | + 0.547·2-s − 1.68·3-s − 0.700·4-s − 0.447·5-s − 0.923·6-s + 1.86·7-s − 0.930·8-s + 1.85·9-s − 0.244·10-s − 0.301·11-s + 1.18·12-s − 1.12·13-s + 1.02·14-s + 0.755·15-s + 0.191·16-s − 1.39·17-s + 1.01·18-s + 1.29·19-s + 0.313·20-s − 3.15·21-s − 0.164·22-s − 1.32·23-s + 1.57·24-s + 0.200·25-s − 0.613·26-s − 1.43·27-s − 1.30·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 - 0.773T + 2T^{2} \) |
| 3 | \( 1 + 2.92T + 3T^{2} \) |
| 7 | \( 1 - 4.94T + 7T^{2} \) |
| 13 | \( 1 + 4.04T + 13T^{2} \) |
| 17 | \( 1 + 5.75T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 6.35T + 23T^{2} \) |
| 29 | \( 1 - 7.49T + 29T^{2} \) |
| 31 | \( 1 - 4.23T + 31T^{2} \) |
| 37 | \( 1 - 5.96T + 37T^{2} \) |
| 41 | \( 1 + 7.85T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 + 4.72T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 1.63T + 59T^{2} \) |
| 61 | \( 1 - 7.97T + 61T^{2} \) |
| 67 | \( 1 - 3.20T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 79 | \( 1 + 2.73T + 79T^{2} \) |
| 83 | \( 1 - 0.334T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−8.123934414364560419924364053939, −7.19928733615434028802558715047, −6.45463579184169771783580652096, −5.48531851963889722933031804522, −4.89778663385125461106834712594, −4.74372406118845917160463528098, −3.94226981135318643407183453803, −2.40321042098565878995136819237, −1.10000999821266047508148284952, 0,
1.10000999821266047508148284952, 2.40321042098565878995136819237, 3.94226981135318643407183453803, 4.74372406118845917160463528098, 4.89778663385125461106834712594, 5.48531851963889722933031804522, 6.45463579184169771783580652096, 7.19928733615434028802558715047, 8.123934414364560419924364053939