| L(s) = 1 | − 4.56·2-s + 12.8·4-s + 1.42·5-s + 3.82·7-s − 22.2·8-s − 6.50·10-s − 18.2·11-s − 17.4·14-s − 1.40·16-s − 93.1·17-s + 74.6·19-s + 18.3·20-s + 83.5·22-s + 7.35·23-s − 122.·25-s + 49.2·28-s − 211.·29-s − 183.·31-s + 184.·32-s + 425.·34-s + 5.44·35-s + 289.·37-s − 340.·38-s − 31.6·40-s − 131.·41-s + 394.·43-s − 235.·44-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 1.60·4-s + 0.127·5-s + 0.206·7-s − 0.982·8-s − 0.205·10-s − 0.501·11-s − 0.333·14-s − 0.0219·16-s − 1.32·17-s + 0.900·19-s + 0.204·20-s + 0.809·22-s + 0.0666·23-s − 0.983·25-s + 0.332·28-s − 1.35·29-s − 1.06·31-s + 1.01·32-s + 2.14·34-s + 0.0262·35-s + 1.28·37-s − 1.45·38-s − 0.125·40-s − 0.500·41-s + 1.39·43-s − 0.806·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6139656109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6139656109\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.56T + 8T^{2} \) |
| 5 | \( 1 - 1.42T + 125T^{2} \) |
| 7 | \( 1 - 3.82T + 343T^{2} \) |
| 11 | \( 1 + 18.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 93.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 74.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 7.35T + 1.21e4T^{2} \) |
| 29 | \( 1 + 211.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 183.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 289.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 131.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 394.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 201.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 16.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 446.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 475.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 252.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 295.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 892.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 170.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 472.T + 9.12e5T^{2} \) |
| 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
−9.272136700889597037292293155388, −8.340382448533222298426338741364, −7.67979768582013882240440573672, −7.06649246687643719991327648073, −6.08925453495243953159323111108, −5.09792883039967392836892052170, −3.88630514127826801531347403207, −2.48848109626337231739424636252, −1.72842937981048589067577668810, −0.48197316318580626180062243822,
0.48197316318580626180062243822, 1.72842937981048589067577668810, 2.48848109626337231739424636252, 3.88630514127826801531347403207, 5.09792883039967392836892052170, 6.08925453495243953159323111108, 7.06649246687643719991327648073, 7.67979768582013882240440573672, 8.340382448533222298426338741364, 9.272136700889597037292293155388
