| L(s) = 1 | − i·2-s − 2i·3-s − 4-s − 2·6-s + i·7-s + i·8-s − 9-s + 3·11-s + 2i·12-s + 14-s + 16-s + 3i·17-s + i·18-s + 6·19-s + 2·21-s − 3i·22-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.15i·3-s − 0.5·4-s − 0.816·6-s + 0.377i·7-s + 0.353i·8-s − 0.333·9-s + 0.904·11-s + 0.577i·12-s + 0.267·14-s + 0.250·16-s + 0.727i·17-s + 0.235i·18-s + 1.37·19-s + 0.436·21-s − 0.639i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.847174187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.847174187\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 13iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
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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
−9.016878666747083129441069379438, −8.199623803560296612358939184017, −7.49469940747098485113384477479, −6.58290769860461660036032458039, −5.94186547462529206669173945191, −4.82798522752783592425277029904, −3.80332166023422941910161875015, −2.76612750318482204677382224436, −1.74917883815201447932811504155, −0.901249338822713350354124349918,
1.10782098922167989188044175594, 3.03803063902427134660923114549, 3.90074513308399965392210705678, 4.62444299874089903314541316567, 5.33841908241949538187458525476, 6.28700624941458639152996448709, 7.19778619434054481492140435772, 7.79810753547288837223010982022, 9.075300239469897935271370439783, 9.269990576951351258150809431634
