L(s) = 1 | − 2-s − 3-s + 4-s + 0.912·5-s + 6-s + 3.48·7-s − 8-s + 9-s − 0.912·10-s − 0.480·11-s − 12-s + 5.22·13-s − 3.48·14-s − 0.912·15-s + 16-s + 6.25·17-s − 18-s − 19-s + 0.912·20-s − 3.48·21-s + 0.480·22-s + 7.18·23-s + 24-s − 4.16·25-s − 5.22·26-s − 27-s + 3.48·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.407·5-s + 0.408·6-s + 1.31·7-s − 0.353·8-s + 0.333·9-s − 0.288·10-s − 0.144·11-s − 0.288·12-s + 1.44·13-s − 0.930·14-s − 0.235·15-s + 0.250·16-s + 1.51·17-s − 0.235·18-s − 0.229·19-s + 0.203·20-s − 0.760·21-s + 0.102·22-s + 1.49·23-s + 0.204·24-s − 0.833·25-s − 1.02·26-s − 0.192·27-s + 0.658·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.924957138\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.924957138\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
good | 5 | \( 1 - 0.912T + 5T^{2} \) |
| 7 | \( 1 - 3.48T + 7T^{2} \) |
| 11 | \( 1 + 0.480T + 11T^{2} \) |
| 13 | \( 1 - 5.22T + 13T^{2} \) |
| 17 | \( 1 - 6.25T + 17T^{2} \) |
| 23 | \( 1 - 7.18T + 23T^{2} \) |
| 29 | \( 1 - 2.36T + 29T^{2} \) |
| 31 | \( 1 - 2.59T + 31T^{2} \) |
| 37 | \( 1 + 3.99T + 37T^{2} \) |
| 41 | \( 1 - 5.30T + 41T^{2} \) |
| 43 | \( 1 - 5.88T + 43T^{2} \) |
| 47 | \( 1 - 1.45T + 47T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 7.68T + 83T^{2} \) |
| 89 | \( 1 - 4.29T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.229262408624231070129625121660, −7.42114472440173427051424530641, −6.78195830446747983922809458490, −5.70100714535140047577874606211, −5.58861358472137787263954775291, −4.54092649858299547626962893402, −3.64128042258239218300794270368, −2.54857098065108725937764034718, −1.42201657527127267504581576758, −0.989421787597128177557098332874,
0.989421787597128177557098332874, 1.42201657527127267504581576758, 2.54857098065108725937764034718, 3.64128042258239218300794270368, 4.54092649858299547626962893402, 5.58861358472137787263954775291, 5.70100714535140047577874606211, 6.78195830446747983922809458490, 7.42114472440173427051424530641, 8.229262408624231070129625121660