L(s) = 1 | − 5.42·2-s − 3·3-s + 21.4·4-s − 16.8·5-s + 16.2·6-s + 7.69·7-s − 72.8·8-s + 9·9-s + 91.3·10-s − 64.2·12-s − 24.8·13-s − 41.7·14-s + 50.5·15-s + 223.·16-s + 15.9·17-s − 48.8·18-s − 15.1·19-s − 360.·20-s − 23.0·21-s + 17.7·23-s + 218.·24-s + 158.·25-s + 134.·26-s − 27·27-s + 164.·28-s + 128.·29-s − 274.·30-s + ⋯ |
L(s) = 1 | − 1.91·2-s − 0.577·3-s + 2.67·4-s − 1.50·5-s + 1.10·6-s + 0.415·7-s − 3.21·8-s + 0.333·9-s + 2.89·10-s − 1.54·12-s − 0.530·13-s − 0.797·14-s + 0.870·15-s + 3.49·16-s + 0.227·17-s − 0.639·18-s − 0.182·19-s − 4.03·20-s − 0.239·21-s + 0.160·23-s + 1.85·24-s + 1.27·25-s + 1.01·26-s − 0.192·27-s + 1.11·28-s + 0.823·29-s − 1.66·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 5.42T + 8T^{2} \) |
| 5 | \( 1 + 16.8T + 125T^{2} \) |
| 7 | \( 1 - 7.69T + 343T^{2} \) |
| 13 | \( 1 + 24.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 15.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 15.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 17.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 128.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 219.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 92.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 459.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 64.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 497.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 526.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 578.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 221.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 860.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 580.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 510.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 606.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 23.4T + 7.04e5T^{2} \) |
| 97 | \( 1 - 719.T + 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.58295092407650123014605870380, −9.612257831573193776172132670074, −8.542911508523817285204120787867, −7.81296683884895451903654150434, −7.23369437708283712577341620555, −6.15240073226701506408318090879, −4.47923496325627273840145314801, −2.82115545269818782078599618056, −1.09710358176811817074855568181, 0,
1.09710358176811817074855568181, 2.82115545269818782078599618056, 4.47923496325627273840145314801, 6.15240073226701506408318090879, 7.23369437708283712577341620555, 7.81296683884895451903654150434, 8.542911508523817285204120787867, 9.612257831573193776172132670074, 10.58295092407650123014605870380