L(s) = 1 | + 2.17·2-s − 3-s + 2.73·4-s − 2.43·5-s − 2.17·6-s − 3.40·7-s + 1.60·8-s + 9-s − 5.30·10-s + 5.76·11-s − 2.73·12-s + 13-s − 7.40·14-s + 2.43·15-s − 1.98·16-s + 7.18·17-s + 2.17·18-s − 1.63·19-s − 6.67·20-s + 3.40·21-s + 12.5·22-s − 3.38·23-s − 1.60·24-s + 0.937·25-s + 2.17·26-s − 27-s − 9.31·28-s + ⋯ |
L(s) = 1 | + 1.53·2-s − 0.577·3-s + 1.36·4-s − 1.08·5-s − 0.888·6-s − 1.28·7-s + 0.567·8-s + 0.333·9-s − 1.67·10-s + 1.73·11-s − 0.790·12-s + 0.277·13-s − 1.98·14-s + 0.629·15-s − 0.495·16-s + 1.74·17-s + 0.513·18-s − 0.375·19-s − 1.49·20-s + 0.742·21-s + 2.67·22-s − 0.706·23-s − 0.327·24-s + 0.187·25-s + 0.426·26-s − 0.192·27-s − 1.76·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 5 | \( 1 + 2.43T + 5T^{2} \) |
| 7 | \( 1 + 3.40T + 7T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 17 | \( 1 - 7.18T + 17T^{2} \) |
| 19 | \( 1 + 1.63T + 19T^{2} \) |
| 23 | \( 1 + 3.38T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 + 8.71T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 8.81T + 41T^{2} \) |
| 43 | \( 1 + 8.64T + 43T^{2} \) |
| 47 | \( 1 + 6.01T + 47T^{2} \) |
| 53 | \( 1 + 5.31T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 9.15T + 61T^{2} \) |
| 67 | \( 1 + 1.53T + 67T^{2} \) |
| 71 | \( 1 + 8.11T + 71T^{2} \) |
| 73 | \( 1 - 6.31T + 73T^{2} \) |
| 79 | \( 1 + 7.98T + 79T^{2} \) |
| 83 | \( 1 - 9.56T + 83T^{2} \) |
| 89 | \( 1 - 4.17T + 89T^{2} \) |
| 97 | \( 1 + 6.07T + 97T^{2} \) |
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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
−7.75953720493660517785268969667, −6.96994563011139733904738374912, −6.33809859856228344712348267028, −5.96245111416169225484823590572, −4.95926193780200131006175526868, −4.13510564748209925722994588827, −3.52424449758857280023193588494, −3.22239247154606494428340955860, −1.51252240294652171275195449362, 0,
1.51252240294652171275195449362, 3.22239247154606494428340955860, 3.52424449758857280023193588494, 4.13510564748209925722994588827, 4.95926193780200131006175526868, 5.96245111416169225484823590572, 6.33809859856228344712348267028, 6.96994563011139733904738374912, 7.75953720493660517785268969667