| L(s) = 1 | − 2.32i·3-s − 1.39i·7-s − 2.39·9-s − 4.32i·13-s + 0.601i·17-s − 19-s − 3.24·21-s − 6.04i·23-s − 1.39i·27-s − 4.60·29-s + 2.79·31-s + 1.07i·37-s − 10.0·39-s − 5.44·41-s + 8.64i·43-s + ⋯ |
| L(s) = 1 | − 1.34i·3-s − 0.528i·7-s − 0.799·9-s − 1.19i·13-s + 0.145i·17-s − 0.229·19-s − 0.708·21-s − 1.26i·23-s − 0.269i·27-s − 0.854·29-s + 0.502·31-s + 0.176i·37-s − 1.60·39-s − 0.850·41-s + 1.31i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.160978111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.160978111\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.32iT - 3T^{2} \) |
| 7 | \( 1 + 1.39iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4.32iT - 13T^{2} \) |
| 17 | \( 1 - 0.601iT - 17T^{2} \) |
| 23 | \( 1 + 6.04iT - 23T^{2} \) |
| 29 | \( 1 + 4.60T + 29T^{2} \) |
| 31 | \( 1 - 2.79T + 31T^{2} \) |
| 37 | \( 1 - 1.07iT - 37T^{2} \) |
| 41 | \( 1 + 5.44T + 41T^{2} \) |
| 43 | \( 1 - 8.64iT - 43T^{2} \) |
| 47 | \( 1 + 1.85iT - 47T^{2} \) |
| 53 | \( 1 - 3.11iT - 53T^{2} \) |
| 59 | \( 1 - 6.69T + 59T^{2} \) |
| 61 | \( 1 + 2.64T + 61T^{2} \) |
| 67 | \( 1 + 14.4iT - 67T^{2} \) |
| 71 | \( 1 + 5.59T + 71T^{2} \) |
| 73 | \( 1 + 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 4.64T + 79T^{2} \) |
| 83 | \( 1 - 1.20iT - 83T^{2} \) |
| 89 | \( 1 + 9.44T + 89T^{2} \) |
| 97 | \( 1 - 4.51iT - 97T^{2} \) |
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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
−7.963064896663921291227471183231, −7.40843104755245891182732277941, −6.61540710152001833228520760373, −6.12045306858802990808457147828, −5.17386682172516514936360606494, −4.23071940256840709222236083897, −3.16697465462797393106678710151, −2.30617675650838253723635086167, −1.27309282765583189636570105611, −0.34400869778399379051097310846,
1.64587361621402961688956936471, 2.71488540180933286473424904123, 3.79803893725793442786446647294, 4.18499771242696888959210147481, 5.21700491204007778372766033472, 5.62427070833614467710536656700, 6.72572131064042491774303449444, 7.39728571894688107367864330328, 8.541826102884353559587103949550, 8.979844754746252746078036013071
