| L(s) = 1 | + i·2-s + 3i·3-s − 4-s − 3·6-s − i·8-s − 6·9-s − 11-s − 3i·12-s + 2i·13-s + 16-s − 7i·17-s − 6i·18-s − 5·19-s − i·22-s − 6i·23-s + 3·24-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 1.73i·3-s − 0.5·4-s − 1.22·6-s − 0.353i·8-s − 2·9-s − 0.301·11-s − 0.866i·12-s + 0.554i·13-s + 0.250·16-s − 1.69i·17-s − 1.41i·18-s − 1.14·19-s − 0.213i·22-s − 1.25i·23-s + 0.612·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2012562760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2012562760\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 - iT \) |
| good | 3 | \( 1 - 3iT - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 7iT - 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 13iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 7iT - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 3iT - 83T^{2} \) |
| 89 | \( 1 - 7T + 89T^{2} \) |
| 97 | \( 1 + 18iT - 97T^{2} \) |
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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
−9.000966820342868989578596781248, −8.790818611242418958286584539239, −7.67557595544474979412347171329, −6.70494542988074724774493996844, −5.84164120198541479031362987727, −4.84875168779007060191989644726, −4.56523500952116029024604501821, −3.56622452256058221211025062499, −2.52782078586486341379761992060, −0.07343599865209760623011349203,
1.37032365811489866772754692917, 2.05492021980020190922706994075, 3.08579888448645632818647788432, 4.13323733935386201006904648718, 5.57112573333003287900307051115, 6.04709913803846023219198329529, 7.05571225588969988773460255718, 7.80338259406801486371003718628, 8.400213925352997832824138565696, 9.095446741134938975361896430860
