L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 4·7-s − 8-s + 9-s + 10-s + 12-s + 13-s + 4·14-s − 15-s + 16-s − 18-s − 19-s − 20-s − 4·21-s + 6·23-s − 24-s + 25-s − 26-s + 27-s − 4·28-s + 3·29-s + 30-s + 8·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.277·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.872·21-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.755·28-s + 0.557·29-s + 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.573522750\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.573522750\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.62457019797482, −14.11214989652025, −13.47742871780286, −13.05152080983498, −12.46548869246722, −12.17395166847822, −11.34683583812406, −10.90291234944797, −10.25738600203521, −9.813107331426677, −9.380256169528623, −8.808651357404452, −8.347323955887027, −7.855878061877253, −6.974309495369434, −6.839449456839207, −6.238707401062290, −5.509554279404629, −4.672156010263866, −3.943126904384749, −3.355188639175831, −2.823044330743638, −2.309808871313106, −1.139580196969833, −0.5479244431792825,
0.5479244431792825, 1.139580196969833, 2.309808871313106, 2.823044330743638, 3.355188639175831, 3.943126904384749, 4.672156010263866, 5.509554279404629, 6.238707401062290, 6.839449456839207, 6.974309495369434, 7.855878061877253, 8.347323955887027, 8.808651357404452, 9.380256169528623, 9.813107331426677, 10.25738600203521, 10.90291234944797, 11.34683583812406, 12.17395166847822, 12.46548869246722, 13.05152080983498, 13.47742871780286, 14.11214989652025, 14.62457019797482