| L(s) = 1 | − 5.13·5-s + 4.02i·7-s − 4.30i·11-s + 18.4·13-s − 23.5·17-s − 21.7i·19-s + 30.7i·23-s + 1.34·25-s + 12.6·29-s − 24.5i·31-s − 20.6i·35-s − 18.2·37-s − 38.0·41-s + 34.9i·43-s − 29.6i·47-s + ⋯ |
| L(s) = 1 | − 1.02·5-s + 0.575i·7-s − 0.391i·11-s + 1.42·13-s − 1.38·17-s − 1.14i·19-s + 1.33i·23-s + 0.0536·25-s + 0.437·29-s − 0.791i·31-s − 0.590i·35-s − 0.492·37-s − 0.929·41-s + 0.813i·43-s − 0.630i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.402184486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.402184486\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5.13T + 25T^{2} \) |
| 7 | \( 1 - 4.02iT - 49T^{2} \) |
| 11 | \( 1 + 4.30iT - 121T^{2} \) |
| 13 | \( 1 - 18.4T + 169T^{2} \) |
| 17 | \( 1 + 23.5T + 289T^{2} \) |
| 19 | \( 1 + 21.7iT - 361T^{2} \) |
| 23 | \( 1 - 30.7iT - 529T^{2} \) |
| 29 | \( 1 - 12.6T + 841T^{2} \) |
| 31 | \( 1 + 24.5iT - 961T^{2} \) |
| 37 | \( 1 + 18.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 38.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 34.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 29.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 39.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 65.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 29.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 11.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 140. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 119.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 9.18iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 113. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 7.88T + 7.92e3T^{2} \) |
| 97 | \( 1 - 55.5T + 9.40e3T^{2} \) |
| 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
−8.726278276607394778629893202431, −8.643200496729989297503637564573, −7.59620775388652393479574819466, −6.75593521227611349346588750515, −5.96827989178731263854484569055, −4.98804533062419053706607739566, −4.00570955492631847625319288684, −3.31790903048030112448826236932, −2.09820652445766158426815065583, −0.62780421035023376824409747903,
0.66791169795716857855030019133, 1.98917915698573643982210310553, 3.44106177573650451902310343962, 4.05031309745877579385149770536, 4.79003550541352815884732748843, 6.09222318036353726668826957897, 6.78771448356279579337904799321, 7.56155705002770759586429198754, 8.502226871058424534105450693641, 8.761824042765521090219963683398
