L(s) = 1 | + 0.311i·2-s − 2.90i·3-s + 1.90·4-s + 0.903·6-s − 4.42i·7-s + 1.21i·8-s − 5.42·9-s − 2.62·11-s − 5.52i·12-s − 0.474i·13-s + 1.37·14-s + 3.42·16-s + 5.05i·17-s − 1.68i·18-s + 19-s + ⋯ |
L(s) = 1 | + 0.219i·2-s − 1.67i·3-s + 0.951·4-s + 0.368·6-s − 1.67i·7-s + 0.429i·8-s − 1.80·9-s − 0.790·11-s − 1.59i·12-s − 0.131i·13-s + 0.368·14-s + 0.857·16-s + 1.22i·17-s − 0.398i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.843435 - 1.36470i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.843435 - 1.36470i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 - 0.311iT - 2T^{2} \) |
| 3 | \( 1 + 2.90iT - 3T^{2} \) |
| 7 | \( 1 + 4.42iT - 7T^{2} \) |
| 11 | \( 1 + 2.62T + 11T^{2} \) |
| 13 | \( 1 + 0.474iT - 13T^{2} \) |
| 17 | \( 1 - 5.05iT - 17T^{2} \) |
| 23 | \( 1 - 1.37iT - 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 - 4.47iT - 37T^{2} \) |
| 41 | \( 1 + 5.05T + 41T^{2} \) |
| 43 | \( 1 + 12.0iT - 43T^{2} \) |
| 47 | \( 1 + 4.42iT - 47T^{2} \) |
| 53 | \( 1 + 7.52iT - 53T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 - 3.67T + 61T^{2} \) |
| 67 | \( 1 + 1.65iT - 67T^{2} \) |
| 71 | \( 1 - 7.61T + 71T^{2} \) |
| 73 | \( 1 - 3.80iT - 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 10.6iT - 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 17.8iT - 97T^{2} \) |
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
−10.71238600759131251005371104697, −10.23806763669370332366678346773, −8.189875333322694638206566343079, −7.926374746338273658165290183645, −6.85114670027400913979392584121, −6.63474283901837241344896074599, −5.33877303064088435315633766897, −3.53717722217027165533086682704, −2.17654260545884217218747931326, −0.996035649149581764977218683675,
2.54237713377928908844146120341, 3.10221209064181506472385403439, 4.70864639395934645839245544882, 5.45141472762195219990169313067, 6.39036593036123170847674868646, 7.895018504508552980281590541580, 8.905978329009433873800971900938, 9.639639164085888516717912494550, 10.38248856497458449943756323914, 11.25991276298869515461305602778