L(s) = 1 | − 2-s − 3-s + 4-s − 1.52·5-s + 6-s − 3.40·7-s − 8-s + 9-s + 1.52·10-s + 2.89·11-s − 12-s + 13-s + 3.40·14-s + 1.52·15-s + 16-s + 3.63·17-s − 18-s + 8.33·19-s − 1.52·20-s + 3.40·21-s − 2.89·22-s − 8.49·23-s + 24-s − 2.68·25-s − 26-s − 27-s − 3.40·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.680·5-s + 0.408·6-s − 1.28·7-s − 0.353·8-s + 0.333·9-s + 0.481·10-s + 0.872·11-s − 0.288·12-s + 0.277·13-s + 0.910·14-s + 0.393·15-s + 0.250·16-s + 0.880·17-s − 0.235·18-s + 1.91·19-s − 0.340·20-s + 0.743·21-s − 0.616·22-s − 1.77·23-s + 0.204·24-s − 0.536·25-s − 0.196·26-s − 0.192·27-s − 0.643·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7475195233\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7475195233\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 + 1.52T + 5T^{2} \) |
| 7 | \( 1 + 3.40T + 7T^{2} \) |
| 11 | \( 1 - 2.89T + 11T^{2} \) |
| 17 | \( 1 - 3.63T + 17T^{2} \) |
| 19 | \( 1 - 8.33T + 19T^{2} \) |
| 23 | \( 1 + 8.49T + 23T^{2} \) |
| 29 | \( 1 + 1.96T + 29T^{2} \) |
| 31 | \( 1 - 2.49T + 31T^{2} \) |
| 37 | \( 1 - 5.76T + 37T^{2} \) |
| 41 | \( 1 - 1.30T + 41T^{2} \) |
| 43 | \( 1 + 6.49T + 43T^{2} \) |
| 47 | \( 1 - 4.75T + 47T^{2} \) |
| 53 | \( 1 - 7.93T + 53T^{2} \) |
| 59 | \( 1 + 7.98T + 59T^{2} \) |
| 61 | \( 1 - 8.78T + 61T^{2} \) |
| 67 | \( 1 - 3.78T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 8.45T + 73T^{2} \) |
| 79 | \( 1 - 3.93T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−7.74255313003321294245922098627, −7.24677405307463966901276924855, −6.45281284390416049510489348230, −5.96818519169025333390077407125, −5.27079222580976633789216686351, −3.91287143156017259403074175054, −3.68158742548686267866313379530, −2.70032855286170308791352631450, −1.39911162262814804442677282062, −0.52975759548789676351445315703,
0.52975759548789676351445315703, 1.39911162262814804442677282062, 2.70032855286170308791352631450, 3.68158742548686267866313379530, 3.91287143156017259403074175054, 5.27079222580976633789216686351, 5.96818519169025333390077407125, 6.45281284390416049510489348230, 7.24677405307463966901276924855, 7.74255313003321294245922098627