| L(s) = 1 | − 1.73·2-s + 3-s + 0.993·4-s − 5-s − 1.73·6-s + 0.0923·7-s + 1.74·8-s + 9-s + 1.73·10-s − 5.90·11-s + 0.993·12-s − 13-s − 0.159·14-s − 15-s − 4.99·16-s − 3.12·17-s − 1.73·18-s + 4.55·19-s − 0.993·20-s + 0.0923·21-s + 10.2·22-s + 0.641·23-s + 1.74·24-s + 25-s + 1.73·26-s + 27-s + 0.0917·28-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.577·3-s + 0.496·4-s − 0.447·5-s − 0.706·6-s + 0.0348·7-s + 0.615·8-s + 0.333·9-s + 0.547·10-s − 1.77·11-s + 0.286·12-s − 0.277·13-s − 0.0426·14-s − 0.258·15-s − 1.24·16-s − 0.758·17-s − 0.407·18-s + 1.04·19-s − 0.222·20-s + 0.0201·21-s + 2.17·22-s + 0.133·23-s + 0.355·24-s + 0.200·25-s + 0.339·26-s + 0.192·27-s + 0.0173·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 7 | \( 1 - 0.0923T + 7T^{2} \) |
| 11 | \( 1 + 5.90T + 11T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 - 4.55T + 19T^{2} \) |
| 23 | \( 1 - 0.641T + 23T^{2} \) |
| 29 | \( 1 - 2.36T + 29T^{2} \) |
| 37 | \( 1 - 3.89T + 37T^{2} \) |
| 41 | \( 1 - 7.91T + 41T^{2} \) |
| 43 | \( 1 - 1.48T + 43T^{2} \) |
| 47 | \( 1 - 1.50T + 47T^{2} \) |
| 53 | \( 1 + 1.30T + 53T^{2} \) |
| 59 | \( 1 - 4.04T + 59T^{2} \) |
| 61 | \( 1 - 9.50T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + 6.45T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 6.56T + 83T^{2} \) |
| 89 | \( 1 - 0.704T + 89T^{2} \) |
| 97 | \( 1 + 7.44T + 97T^{2} \) |
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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
−7.988883583944986664326757731989, −7.36789035123636097330100031636, −6.73970657323738920328497770544, −5.49041950985195213508940487319, −4.80487953168403556695789958934, −4.03727792781893023919602484239, −2.85071577993287744569872559046, −2.33409284168058953238980114135, −1.07473471152057436628654135048, 0,
1.07473471152057436628654135048, 2.33409284168058953238980114135, 2.85071577993287744569872559046, 4.03727792781893023919602484239, 4.80487953168403556695789958934, 5.49041950985195213508940487319, 6.73970657323738920328497770544, 7.36789035123636097330100031636, 7.988883583944986664326757731989
