| L(s) = 1 | − 0.571·2-s + 3-s − 1.67·4-s − 2.67·5-s − 0.571·6-s − 3.67·7-s + 2.10·8-s + 9-s + 1.52·10-s − 3.81·11-s − 1.67·12-s + 0.143·13-s + 2.10·14-s − 2.67·15-s + 2.14·16-s − 0.571·18-s + 4.47·20-s − 3.67·21-s + 2.18·22-s + 7.52·23-s + 2.10·24-s + 2.14·25-s − 0.0823·26-s + 27-s + 6.14·28-s − 5.34·29-s + 1.52·30-s + ⋯ |
| L(s) = 1 | − 0.404·2-s + 0.577·3-s − 0.836·4-s − 1.19·5-s − 0.233·6-s − 1.38·7-s + 0.742·8-s + 0.333·9-s + 0.483·10-s − 1.15·11-s − 0.482·12-s + 0.0399·13-s + 0.561·14-s − 0.690·15-s + 0.535·16-s − 0.134·18-s + 0.999·20-s − 0.801·21-s + 0.465·22-s + 1.56·23-s + 0.428·24-s + 0.428·25-s − 0.0161·26-s + 0.192·27-s + 1.16·28-s − 0.992·29-s + 0.279·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6192455677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6192455677\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.571T + 2T^{2} \) |
| 5 | \( 1 + 2.67T + 5T^{2} \) |
| 7 | \( 1 + 3.67T + 7T^{2} \) |
| 11 | \( 1 + 3.81T + 11T^{2} \) |
| 13 | \( 1 - 0.143T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 - 7.52T + 23T^{2} \) |
| 29 | \( 1 + 5.34T + 29T^{2} \) |
| 31 | \( 1 - 8.81T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 5.34T + 41T^{2} \) |
| 43 | \( 1 - 2.81T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 8.01T + 53T^{2} \) |
| 59 | \( 1 - 3.81T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 5.38T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 0.345T + 73T^{2} \) |
| 79 | \( 1 - 6.52T + 79T^{2} \) |
| 83 | \( 1 + 2.28T + 83T^{2} \) |
| 89 | \( 1 + 8.67T + 89T^{2} \) |
| 97 | \( 1 + 5.91T + 97T^{2} \) |
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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
−9.772177140935082487427771238511, −9.000219442929019549236696144475, −8.283619430828854722938608339671, −7.59236858195974866173836410043, −6.85775037422429070383656355746, −5.48178989383709369810953620445, −4.43637026507031162514883286637, −3.58097497061188803462347285515, −2.77759438759179916673741687981, −0.60456123115622303950618124783,
0.60456123115622303950618124783, 2.77759438759179916673741687981, 3.58097497061188803462347285515, 4.43637026507031162514883286637, 5.48178989383709369810953620445, 6.85775037422429070383656355746, 7.59236858195974866173836410043, 8.283619430828854722938608339671, 9.000219442929019549236696144475, 9.772177140935082487427771238511
