| L(s) = 1 | + 2.46·2-s − 3-s + 4.09·4-s + 5-s − 2.46·6-s − 7-s + 5.15·8-s + 9-s + 2.46·10-s + 5.09·11-s − 4.09·12-s − 0.461·13-s − 2.46·14-s − 15-s + 4.55·16-s − 17-s + 2.46·18-s + 2·19-s + 4.09·20-s + 21-s + 12.5·22-s − 1.15·23-s − 5.15·24-s + 25-s − 1.13·26-s − 27-s − 4.09·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s − 0.577·3-s + 2.04·4-s + 0.447·5-s − 1.00·6-s − 0.377·7-s + 1.82·8-s + 0.333·9-s + 0.780·10-s + 1.53·11-s − 1.18·12-s − 0.128·13-s − 0.659·14-s − 0.258·15-s + 1.13·16-s − 0.242·17-s + 0.581·18-s + 0.458·19-s + 0.914·20-s + 0.218·21-s + 2.67·22-s − 0.240·23-s − 1.05·24-s + 0.200·25-s − 0.223·26-s − 0.192·27-s − 0.773·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.700790578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.700790578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 11 | \( 1 - 5.09T + 11T^{2} \) |
| 13 | \( 1 + 0.461T + 13T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 1.15T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 - 4.29T + 31T^{2} \) |
| 37 | \( 1 + 2.23T + 37T^{2} \) |
| 41 | \( 1 + 4.30T + 41T^{2} \) |
| 43 | \( 1 - 5.33T + 43T^{2} \) |
| 47 | \( 1 + 7.38T + 47T^{2} \) |
| 53 | \( 1 + 0.691T + 53T^{2} \) |
| 59 | \( 1 + 3.73T + 59T^{2} \) |
| 61 | \( 1 - 8.96T + 61T^{2} \) |
| 67 | \( 1 - 3.15T + 67T^{2} \) |
| 71 | \( 1 - 8.29T + 71T^{2} \) |
| 73 | \( 1 - 2.70T + 73T^{2} \) |
| 79 | \( 1 + 4.93T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 8.58T + 89T^{2} \) |
| 97 | \( 1 + 6.58T + 97T^{2} \) |
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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
−9.511872168624075647253447211564, −8.434315458376606741783970360455, −7.05072153429717060336635983800, −6.57401551005486748842532159933, −6.02973105513635434249178378518, −5.14594943860019218434763631113, −4.39933390822749981525643999541, −3.61534940514463236847955847788, −2.61666086751245302536262113995, −1.35489160776684380839810096408,
1.35489160776684380839810096408, 2.61666086751245302536262113995, 3.61534940514463236847955847788, 4.39933390822749981525643999541, 5.14594943860019218434763631113, 6.02973105513635434249178378518, 6.57401551005486748842532159933, 7.05072153429717060336635983800, 8.434315458376606741783970360455, 9.511872168624075647253447211564
