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 | \( 1 + 17.8iT - 97T^{2} \) |
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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
−11.25991276298869515461305602778, −10.38248856497458449943756323914, −9.639639164085888516717912494550, −8.905978329009433873800971900938, −7.895018504508552980281590541580, −6.39036593036123170847674868646, −5.45141472762195219990169313067, −4.70864639395934645839245544882, −3.10221209064181506472385403439, −2.54237713377928908844146120341,
0.996035649149581764977218683675, 2.17654260545884217218747931326, 3.53717722217027165533086682704, 5.33877303064088435315633766897, 6.63474283901837241344896074599, 6.85114670027400913979392584121, 7.926374746338273658165290183645, 8.189875333322694638206566343079, 10.23806763669370332366678346773, 10.71238600759131251005371104697