L(s) = 1 | − 2.23·2-s − 3-s + 2.97·4-s + 0.320·5-s + 2.23·6-s + 0.645·7-s − 2.18·8-s + 9-s − 0.714·10-s + 3.13·11-s − 2.97·12-s + 1.44·13-s − 1.44·14-s − 0.320·15-s − 1.08·16-s + 17-s − 2.23·18-s + 5.27·19-s + 0.954·20-s − 0.645·21-s − 6.99·22-s − 1.47·23-s + 2.18·24-s − 4.89·25-s − 3.21·26-s − 27-s + 1.92·28-s + ⋯ |
L(s) = 1 | − 1.57·2-s − 0.577·3-s + 1.48·4-s + 0.143·5-s + 0.910·6-s + 0.244·7-s − 0.770·8-s + 0.333·9-s − 0.226·10-s + 0.945·11-s − 0.859·12-s + 0.400·13-s − 0.384·14-s − 0.0827·15-s − 0.272·16-s + 0.242·17-s − 0.525·18-s + 1.21·19-s + 0.213·20-s − 0.140·21-s − 1.49·22-s − 0.308·23-s + 0.445·24-s − 0.979·25-s − 0.631·26-s − 0.192·27-s + 0.363·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 5 | \( 1 - 0.320T + 5T^{2} \) |
| 7 | \( 1 - 0.645T + 7T^{2} \) |
| 11 | \( 1 - 3.13T + 11T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 19 | \( 1 - 5.27T + 19T^{2} \) |
| 23 | \( 1 + 1.47T + 23T^{2} \) |
| 29 | \( 1 + 0.913T + 29T^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 37 | \( 1 + 1.82T + 37T^{2} \) |
| 41 | \( 1 - 5.92T + 41T^{2} \) |
| 43 | \( 1 + 4.59T + 43T^{2} \) |
| 47 | \( 1 + 3.20T + 47T^{2} \) |
| 53 | \( 1 - 1.53T + 53T^{2} \) |
| 59 | \( 1 + 7.87T + 59T^{2} \) |
| 61 | \( 1 + 2.43T + 61T^{2} \) |
| 67 | \( 1 + 5.18T + 67T^{2} \) |
| 71 | \( 1 + 3.67T + 71T^{2} \) |
| 73 | \( 1 + 6.02T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 2.81T + 83T^{2} \) |
| 89 | \( 1 - 8.86T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−7.64381607589815362081973281877, −6.97077751545582078293099337388, −6.29429100057072010161654849027, −5.68459995170058247052481990582, −4.71526927498589144624375403631, −3.88606973201737231943069549619, −2.84266703104016189645387572848, −1.61370730758379306410207566668, −1.22117711495811691475500745188, 0,
1.22117711495811691475500745188, 1.61370730758379306410207566668, 2.84266703104016189645387572848, 3.88606973201737231943069549619, 4.71526927498589144624375403631, 5.68459995170058247052481990582, 6.29429100057072010161654849027, 6.97077751545582078293099337388, 7.64381607589815362081973281877