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 | \( 1 - 10iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.453679605485522263187435528469, −8.281028368073281694480300016272, −6.74817882688283841819901377021, −6.53160607052570787030599976959, −5.21848510036479502162006803669, −4.37144490027030198185244195611, −4.02687175180789504037043217108, −3.00778413161477168460644940239, −1.99294127676939544676961041320, −0.77961067081901145361420462777,
0.74854409572395087390692142387, 2.11006488667088853878591070086, 2.99801721752752521504811295188, 4.21041166663859231026408996133, 4.92166332292426082433719553744, 5.97046187158078084221185552061, 6.35620876400494674680637922228, 7.01693371565295696275902382588, 8.049624222480126240166021194579, 8.491725685986546108368650146300