| L(s) = 1 | + 1.49·2-s − 5.77·4-s − 16.6·5-s − 17.5·7-s − 20.5·8-s − 24.7·10-s − 31.0·11-s + 18.6·13-s − 26.2·14-s + 15.5·16-s + 39.8·17-s + 77.6·19-s + 96.0·20-s − 46.2·22-s + 67.5·23-s + 151.·25-s + 27.7·26-s + 101.·28-s + 255.·29-s − 191.·31-s + 187.·32-s + 59.4·34-s + 292.·35-s − 320.·37-s + 115.·38-s + 341.·40-s + 475.·41-s + ⋯ |
| L(s) = 1 | + 0.527·2-s − 0.722·4-s − 1.48·5-s − 0.949·7-s − 0.907·8-s − 0.784·10-s − 0.849·11-s + 0.397·13-s − 0.500·14-s + 0.243·16-s + 0.569·17-s + 0.937·19-s + 1.07·20-s − 0.447·22-s + 0.612·23-s + 1.21·25-s + 0.209·26-s + 0.685·28-s + 1.63·29-s − 1.11·31-s + 1.03·32-s + 0.300·34-s + 1.41·35-s − 1.42·37-s + 0.494·38-s + 1.35·40-s + 1.81·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 - 1.49T + 8T^{2} \) |
| 5 | \( 1 + 16.6T + 125T^{2} \) |
| 7 | \( 1 + 17.5T + 343T^{2} \) |
| 11 | \( 1 + 31.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 39.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 77.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 67.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 255.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 191.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 320.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 475.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 7.04T + 7.95e4T^{2} \) |
| 47 | \( 1 + 182.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 121.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 374.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 445.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 391.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 139.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 440.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 132.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 222.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 32.3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 780.T + 9.12e5T^{2} \) |
| 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
−8.313094294224143649279460766860, −7.60109582815160726887180093293, −6.83634708468688174696811189259, −5.76575088744441738550616233515, −5.03583917963455324770233773043, −4.18207941944151313955924769779, −3.37603517219752671833477325127, −2.96153611398685003300224954207, −0.857119604363712245824395233519, 0,
0.857119604363712245824395233519, 2.96153611398685003300224954207, 3.37603517219752671833477325127, 4.18207941944151313955924769779, 5.03583917963455324770233773043, 5.76575088744441738550616233515, 6.83634708468688174696811189259, 7.60109582815160726887180093293, 8.313094294224143649279460766860
