| L(s) = 1 | − 2-s + 3-s + 4-s − 0.757·5-s − 6-s + 1.64·7-s − 8-s + 9-s + 0.757·10-s + 1.41·11-s + 12-s + 13-s − 1.64·14-s − 0.757·15-s + 16-s − 1.39·17-s − 18-s + 8.50·19-s − 0.757·20-s + 1.64·21-s − 1.41·22-s + 2.96·23-s − 24-s − 4.42·25-s − 26-s + 27-s + 1.64·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.338·5-s − 0.408·6-s + 0.622·7-s − 0.353·8-s + 0.333·9-s + 0.239·10-s + 0.426·11-s + 0.288·12-s + 0.277·13-s − 0.439·14-s − 0.195·15-s + 0.250·16-s − 0.337·17-s − 0.235·18-s + 1.95·19-s − 0.169·20-s + 0.359·21-s − 0.301·22-s + 0.618·23-s − 0.204·24-s − 0.885·25-s − 0.196·26-s + 0.192·27-s + 0.311·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.188780909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.188780909\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 + 0.757T + 5T^{2} \) |
| 7 | \( 1 - 1.64T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 17 | \( 1 + 1.39T + 17T^{2} \) |
| 19 | \( 1 - 8.50T + 19T^{2} \) |
| 23 | \( 1 - 2.96T + 23T^{2} \) |
| 29 | \( 1 - 8.43T + 29T^{2} \) |
| 31 | \( 1 - 0.646T + 31T^{2} \) |
| 37 | \( 1 + 8.78T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 + 0.841T + 43T^{2} \) |
| 47 | \( 1 - 2.52T + 47T^{2} \) |
| 53 | \( 1 - 5.66T + 53T^{2} \) |
| 59 | \( 1 - 2.01T + 59T^{2} \) |
| 61 | \( 1 + 5.39T + 61T^{2} \) |
| 67 | \( 1 - 8.33T + 67T^{2} \) |
| 71 | \( 1 + 3.32T + 71T^{2} \) |
| 73 | \( 1 + 6.98T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 + 3.92T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 + 6.37T + 97T^{2} \) |
| 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
−7.83391892103477189256967209248, −7.38136927160007425969320212978, −6.72763902435759589159134097887, −5.82277663025634595053237100585, −5.01247168694560628032667700725, −4.18764038639208689341265149012, −3.35348632540663162872639821152, −2.64421362116125853427877375991, −1.59268106663676917219662166721, −0.855037885004529546812495439673,
0.855037885004529546812495439673, 1.59268106663676917219662166721, 2.64421362116125853427877375991, 3.35348632540663162872639821152, 4.18764038639208689341265149012, 5.01247168694560628032667700725, 5.82277663025634595053237100585, 6.72763902435759589159134097887, 7.38136927160007425969320212978, 7.83391892103477189256967209248
