| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s + 2i·7-s − i·8-s − 9-s + 2·11-s + i·12-s − 2i·13-s − 2·14-s + 16-s − i·18-s − 8·19-s + 2·21-s + 2i·22-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.755i·7-s − 0.353i·8-s − 0.333·9-s + 0.603·11-s + 0.288i·12-s − 0.554i·13-s − 0.534·14-s + 0.250·16-s − 0.235i·18-s − 1.83·19-s + 0.436·21-s + 0.426i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.650111563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.650111563\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 10iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
| 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
−8.491725685986546108368650146300, −8.049624222480126240166021194579, −7.01693371565295696275902382588, −6.35620876400494674680637922228, −5.97046187158078084221185552061, −4.92166332292426082433719553744, −4.21041166663859231026408996133, −2.99801721752752521504811295188, −2.11006488667088853878591070086, −0.74854409572395087390692142387,
0.77961067081901145361420462777, 1.99294127676939544676961041320, 3.00778413161477168460644940239, 4.02687175180789504037043217108, 4.37144490027030198185244195611, 5.21848510036479502162006803669, 6.53160607052570787030599976959, 6.74817882688283841819901377021, 8.281028368073281694480300016272, 8.453679605485522263187435528469
