| L(s) = 1 | + 2.23i·2-s − i·3-s − 2.98·4-s + 2.23·6-s − 1.03i·7-s − 2.20i·8-s − 9-s + 6.17·11-s + 2.98i·12-s − 0.937i·13-s + 2.30·14-s − 1.04·16-s − 6.56i·17-s − 2.23i·18-s − 5.67·19-s + ⋯ |
| L(s) = 1 | + 1.57i·2-s − 0.577i·3-s − 1.49·4-s + 0.911·6-s − 0.389i·7-s − 0.781i·8-s − 0.333·9-s + 1.86·11-s + 0.862i·12-s − 0.260i·13-s + 0.615·14-s − 0.260·16-s − 1.59i·17-s − 0.526i·18-s − 1.30·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.358862957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.358862957\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.23iT - 2T^{2} \) |
| 7 | \( 1 + 1.03iT - 7T^{2} \) |
| 11 | \( 1 - 6.17T + 11T^{2} \) |
| 13 | \( 1 + 0.937iT - 13T^{2} \) |
| 17 | \( 1 + 6.56iT - 17T^{2} \) |
| 19 | \( 1 + 5.67T + 19T^{2} \) |
| 23 | \( 1 + 1.64iT - 23T^{2} \) |
| 29 | \( 1 + 8.35T + 29T^{2} \) |
| 31 | \( 1 - 5.53T + 31T^{2} \) |
| 37 | \( 1 + 1.29iT - 37T^{2} \) |
| 41 | \( 1 + 4.98T + 41T^{2} \) |
| 43 | \( 1 + 7.75iT - 43T^{2} \) |
| 47 | \( 1 + 7.67iT - 47T^{2} \) |
| 53 | \( 1 + 0.500iT - 53T^{2} \) |
| 59 | \( 1 - 1.19T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 + 7.58iT - 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 7.98iT - 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 1.46iT - 83T^{2} \) |
| 89 | \( 1 - 8.51T + 89T^{2} \) |
| 97 | \( 1 - 3.75iT - 97T^{2} \) |
| 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
−8.983563064613826902007179919292, −8.284044943154871899678305451689, −7.39059089439775798536448002374, −6.83180439632130766676284877728, −6.37914048446029676838361946286, −5.46351214594692694868516574040, −4.54003495469921900136954221296, −3.66757102868796213074272555449, −2.08677210981410383909640662411, −0.52612258338638026373027927121,
1.35630087567831747861071514788, 2.15464902516338168425182354767, 3.43477687909764237196107007007, 3.99887590125730691411683465721, 4.62642877347327726221055332083, 6.02945540157206424763461764686, 6.58372188172104840769552077138, 8.078960661655287815080801971008, 8.988634439572211622795327220007, 9.265636644999270397756257812415
