| L(s) = 1 | − 2.46·2-s + 2.34·3-s − 1.93·4-s + 7.40·5-s − 5.77·6-s − 12.1·7-s + 24.4·8-s − 21.4·9-s − 18.2·10-s + 5.10·11-s − 4.54·12-s + 24.2·13-s + 30.0·14-s + 17.3·15-s − 44.7·16-s + 64.3·17-s + 52.9·18-s − 49.5·19-s − 14.3·20-s − 28.5·21-s − 12.5·22-s + 57.4·24-s − 70.2·25-s − 59.7·26-s − 113.·27-s + 23.6·28-s + 172.·29-s + ⋯ |
| L(s) = 1 | − 0.870·2-s + 0.451·3-s − 0.242·4-s + 0.662·5-s − 0.393·6-s − 0.657·7-s + 1.08·8-s − 0.796·9-s − 0.576·10-s + 0.139·11-s − 0.109·12-s + 0.517·13-s + 0.572·14-s + 0.299·15-s − 0.699·16-s + 0.918·17-s + 0.693·18-s − 0.598·19-s − 0.160·20-s − 0.297·21-s − 0.121·22-s + 0.488·24-s − 0.561·25-s − 0.450·26-s − 0.811·27-s + 0.159·28-s + 1.10·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 \) |
| good | 2 | \( 1 + 2.46T + 8T^{2} \) |
| 3 | \( 1 - 2.34T + 27T^{2} \) |
| 5 | \( 1 - 7.40T + 125T^{2} \) |
| 7 | \( 1 + 12.1T + 343T^{2} \) |
| 11 | \( 1 - 5.10T + 1.33e3T^{2} \) |
| 13 | \( 1 - 24.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 64.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 49.5T + 6.85e3T^{2} \) |
| 29 | \( 1 - 172.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 232.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 238.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 311.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 525.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 407.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 481.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 42.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 340.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 530.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 746.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 302.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 775.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 293.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 569.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.86e3T + 9.12e5T^{2} \) |
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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
−9.775337188318628416070273164675, −9.154707445003195308348035471906, −8.420005726539969225669772846547, −7.61024186663524391768333041689, −6.34172472983988603741631577251, −5.48307465244570862327101200385, −4.05190560946023827246089700965, −2.85731733542217983503016875482, −1.48076382389133998732451851631, 0,
1.48076382389133998732451851631, 2.85731733542217983503016875482, 4.05190560946023827246089700965, 5.48307465244570862327101200385, 6.34172472983988603741631577251, 7.61024186663524391768333041689, 8.420005726539969225669772846547, 9.154707445003195308348035471906, 9.775337188318628416070273164675
