| L(s) = 1 | − 4.47·3-s + 5·5-s − 31.3·7-s − 6.99·9-s + 8.94·11-s + 62·13-s − 22.3·15-s − 46·17-s − 107.·19-s + 140·21-s + 192.·23-s + 25·25-s + 152.·27-s + 90·29-s − 152.·31-s − 40.0·33-s − 156.·35-s + 214·37-s − 277.·39-s − 10·41-s + 67.0·43-s − 34.9·45-s + 398.·47-s + 637.·49-s + 205.·51-s + 678·53-s + 44.7·55-s + ⋯ |
| L(s) = 1 | − 0.860·3-s + 0.447·5-s − 1.69·7-s − 0.259·9-s + 0.245·11-s + 1.32·13-s − 0.384·15-s − 0.656·17-s − 1.29·19-s + 1.45·21-s + 1.74·23-s + 0.200·25-s + 1.08·27-s + 0.576·29-s − 0.880·31-s − 0.211·33-s − 0.755·35-s + 0.950·37-s − 1.13·39-s − 0.0380·41-s + 0.237·43-s − 0.115·45-s + 1.23·47-s + 1.85·49-s + 0.564·51-s + 1.75·53-s + 0.109·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.017873778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.017873778\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 3 | \( 1 + 4.47T + 27T^{2} \) |
| 7 | \( 1 + 31.3T + 343T^{2} \) |
| 11 | \( 1 - 8.94T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62T + 2.19e3T^{2} \) |
| 17 | \( 1 + 46T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 192.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 90T + 2.43e4T^{2} \) |
| 31 | \( 1 + 152.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 214T + 5.06e4T^{2} \) |
| 41 | \( 1 + 10T + 6.89e4T^{2} \) |
| 43 | \( 1 - 67.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 398.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 678T + 1.48e5T^{2} \) |
| 59 | \( 1 - 411.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 250T + 2.26e5T^{2} \) |
| 67 | \( 1 + 49.1T + 3.00e5T^{2} \) |
| 71 | \( 1 + 366.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 522T + 3.89e5T^{2} \) |
| 79 | \( 1 - 876.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 380.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 970T + 7.04e5T^{2} \) |
| 97 | \( 1 + 934T + 9.12e5T^{2} \) |
| 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
−10.97748638155270677193538425627, −10.55286502382544894158109623122, −9.261281330206464793554102825460, −8.699027344427225797490265840023, −6.82003109683543425815133556034, −6.35009909812789073548131244174, −5.52194766478119576004314393366, −3.99666622616855055289384092020, −2.72439308021699582370502183569, −0.70740526746535172443980076197,
0.70740526746535172443980076197, 2.72439308021699582370502183569, 3.99666622616855055289384092020, 5.52194766478119576004314393366, 6.35009909812789073548131244174, 6.82003109683543425815133556034, 8.699027344427225797490265840023, 9.261281330206464793554102825460, 10.55286502382544894158109623122, 10.97748638155270677193538425627
