L(s) = 1 | − 2-s + 3-s + 4-s + 1.45·5-s − 6-s + 1.71·7-s − 8-s + 9-s − 1.45·10-s − 2.01·11-s + 12-s − 3.15·13-s − 1.71·14-s + 1.45·15-s + 16-s − 8.17·17-s − 18-s − 19-s + 1.45·20-s + 1.71·21-s + 2.01·22-s + 7.84·23-s − 24-s − 2.87·25-s + 3.15·26-s + 27-s + 1.71·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.651·5-s − 0.408·6-s + 0.648·7-s − 0.353·8-s + 0.333·9-s − 0.460·10-s − 0.607·11-s + 0.288·12-s − 0.875·13-s − 0.458·14-s + 0.376·15-s + 0.250·16-s − 1.98·17-s − 0.235·18-s − 0.229·19-s + 0.325·20-s + 0.374·21-s + 0.429·22-s + 1.63·23-s − 0.204·24-s − 0.575·25-s + 0.618·26-s + 0.192·27-s + 0.324·28-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
good | 5 | \( 1 - 1.45T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 11 | \( 1 + 2.01T + 11T^{2} \) |
| 13 | \( 1 + 3.15T + 13T^{2} \) |
| 17 | \( 1 + 8.17T + 17T^{2} \) |
| 23 | \( 1 - 7.84T + 23T^{2} \) |
| 29 | \( 1 + 5.97T + 29T^{2} \) |
| 31 | \( 1 + 0.954T + 31T^{2} \) |
| 37 | \( 1 + 4.94T + 37T^{2} \) |
| 41 | \( 1 - 7.49T + 41T^{2} \) |
| 43 | \( 1 - 4.28T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 59 | \( 1 + 5.14T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 - 1.93T + 67T^{2} \) |
| 71 | \( 1 - 0.646T + 71T^{2} \) |
| 73 | \( 1 + 0.474T + 73T^{2} \) |
| 79 | \( 1 + 5.19T + 79T^{2} \) |
| 83 | \( 1 + 8.68T + 83T^{2} \) |
| 89 | \( 1 + 0.797T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−7.57723634757156383189254336451, −7.36298732971287027766895761266, −6.48013684348861490500807065928, −5.60622480361230343239046080359, −4.83926954470733324511720246211, −4.08708525500247518145974816924, −2.75952253567178543432588418880, −2.30240067682235730890283366394, −1.49878654115060734478017127134, 0,
1.49878654115060734478017127134, 2.30240067682235730890283366394, 2.75952253567178543432588418880, 4.08708525500247518145974816924, 4.83926954470733324511720246211, 5.60622480361230343239046080359, 6.48013684348861490500807065928, 7.36298732971287027766895761266, 7.57723634757156383189254336451