L(s) = 1 | + 1.66·2-s + 3-s + 0.759·4-s − 1.47·5-s + 1.66·6-s − 2.81·7-s − 2.06·8-s + 9-s − 2.44·10-s + 11-s + 0.759·12-s + 4.11·13-s − 4.67·14-s − 1.47·15-s − 4.94·16-s − 4.46·17-s + 1.66·18-s + 2.28·19-s − 1.11·20-s − 2.81·21-s + 1.66·22-s − 7.49·23-s − 2.06·24-s − 2.82·25-s + 6.84·26-s + 27-s − 2.13·28-s + ⋯ |
L(s) = 1 | + 1.17·2-s + 0.577·3-s + 0.379·4-s − 0.658·5-s + 0.678·6-s − 1.06·7-s − 0.728·8-s + 0.333·9-s − 0.773·10-s + 0.301·11-s + 0.219·12-s + 1.14·13-s − 1.24·14-s − 0.380·15-s − 1.23·16-s − 1.08·17-s + 0.391·18-s + 0.524·19-s − 0.250·20-s − 0.614·21-s + 0.354·22-s − 1.56·23-s − 0.420·24-s − 0.565·25-s + 1.34·26-s + 0.192·27-s − 0.403·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 - 1.66T + 2T^{2} \) |
| 5 | \( 1 + 1.47T + 5T^{2} \) |
| 7 | \( 1 + 2.81T + 7T^{2} \) |
| 13 | \( 1 - 4.11T + 13T^{2} \) |
| 17 | \( 1 + 4.46T + 17T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 23 | \( 1 + 7.49T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 - 4.99T + 31T^{2} \) |
| 37 | \( 1 + 0.685T + 37T^{2} \) |
| 41 | \( 1 + 9.33T + 41T^{2} \) |
| 43 | \( 1 + 6.98T + 43T^{2} \) |
| 47 | \( 1 - 1.45T + 47T^{2} \) |
| 53 | \( 1 + 4.15T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 67 | \( 1 + 2.34T + 67T^{2} \) |
| 71 | \( 1 - 1.88T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 1.63T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.717189910476409699845966509171, −8.048464461013014421088047581063, −6.91565725001030871056076210205, −6.32254460965103473315348613981, −5.54409985348164758754597996065, −4.35347433541129607867744803484, −3.71014142998344576990415956335, −3.31563521952824778801292416974, −2.03514172558212716986341293269, 0,
2.03514172558212716986341293269, 3.31563521952824778801292416974, 3.71014142998344576990415956335, 4.35347433541129607867744803484, 5.54409985348164758754597996065, 6.32254460965103473315348613981, 6.91565725001030871056076210205, 8.048464461013014421088047581063, 8.717189910476409699845966509171