| L(s) = 1 | − 3.13·2-s + 4.96·3-s + 1.85·4-s − 20.3·5-s − 15.5·6-s − 8.13·7-s + 19.2·8-s − 2.35·9-s + 63.9·10-s + 9.18·12-s − 13·13-s + 25.5·14-s − 101.·15-s − 75.3·16-s + 20.7·17-s + 7.40·18-s + 33.3·19-s − 37.7·20-s − 40.3·21-s + 44.7·23-s + 95.8·24-s + 290.·25-s + 40.8·26-s − 145.·27-s − 15.0·28-s − 97.3·29-s + 317.·30-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.955·3-s + 0.231·4-s − 1.82·5-s − 1.06·6-s − 0.439·7-s + 0.852·8-s − 0.0873·9-s + 2.02·10-s + 0.221·12-s − 0.277·13-s + 0.487·14-s − 1.74·15-s − 1.17·16-s + 0.296·17-s + 0.0969·18-s + 0.403·19-s − 0.421·20-s − 0.419·21-s + 0.405·23-s + 0.814·24-s + 2.32·25-s + 0.307·26-s − 1.03·27-s − 0.101·28-s − 0.623·29-s + 1.93·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 3.13T + 8T^{2} \) |
| 3 | \( 1 - 4.96T + 27T^{2} \) |
| 5 | \( 1 + 20.3T + 125T^{2} \) |
| 7 | \( 1 + 8.13T + 343T^{2} \) |
| 17 | \( 1 - 20.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 33.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 44.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 97.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 68.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 343.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 334.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 92.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 89.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 541.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 305.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 69.4T + 2.26e5T^{2} \) |
| 67 | \( 1 - 881.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 427.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 453.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 720.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.43e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 578.T + 9.12e5T^{2} \) |
| 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
−8.445446955030790563194971501111, −8.133152654833584931326791997578, −7.46868782598439345896380480611, −6.80960678045245941950414422937, −5.20056485724964252615966289716, −4.12177588147364479179884361423, −3.48288934294877407951614708289, −2.50136690733754724694684271247, −0.928747757726143021149760235920, 0,
0.928747757726143021149760235920, 2.50136690733754724694684271247, 3.48288934294877407951614708289, 4.12177588147364479179884361423, 5.20056485724964252615966289716, 6.80960678045245941950414422937, 7.46868782598439345896380480611, 8.133152654833584931326791997578, 8.445446955030790563194971501111
