L(s) = 1 | − 3.04i·5-s − 7-s + 0.128i·11-s + 6.30i·13-s + 5.32·17-s + 6.68i·19-s − 5.18·23-s − 4.26·25-s − 9.96i·29-s − 3.27·31-s + 3.04i·35-s − 0.796i·37-s − 2.96·41-s − 6.99i·43-s − 4.76·47-s + ⋯ |
L(s) = 1 | − 1.36i·5-s − 0.377·7-s + 0.0387i·11-s + 1.74i·13-s + 1.29·17-s + 1.53i·19-s − 1.08·23-s − 0.852·25-s − 1.85i·29-s − 0.587·31-s + 0.514i·35-s − 0.131i·37-s − 0.463·41-s − 1.06i·43-s − 0.695·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6363884912\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6363884912\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 3.04iT - 5T^{2} \) |
| 11 | \( 1 - 0.128iT - 11T^{2} \) |
| 13 | \( 1 - 6.30iT - 13T^{2} \) |
| 17 | \( 1 - 5.32T + 17T^{2} \) |
| 19 | \( 1 - 6.68iT - 19T^{2} \) |
| 23 | \( 1 + 5.18T + 23T^{2} \) |
| 29 | \( 1 + 9.96iT - 29T^{2} \) |
| 31 | \( 1 + 3.27T + 31T^{2} \) |
| 37 | \( 1 + 0.796iT - 37T^{2} \) |
| 41 | \( 1 + 2.96T + 41T^{2} \) |
| 43 | \( 1 + 6.99iT - 43T^{2} \) |
| 47 | \( 1 + 4.76T + 47T^{2} \) |
| 53 | \( 1 - 1.14iT - 53T^{2} \) |
| 59 | \( 1 + 11.0iT - 59T^{2} \) |
| 61 | \( 1 + 14.6iT - 61T^{2} \) |
| 67 | \( 1 - 1.05iT - 67T^{2} \) |
| 71 | \( 1 + 1.10T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 2.62T + 79T^{2} \) |
| 83 | \( 1 - 6.46iT - 83T^{2} \) |
| 89 | \( 1 + 2.23T + 89T^{2} \) |
| 97 | \( 1 + 7.88T + 97T^{2} \) |
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
−7.985171488897973483780745965028, −7.08775815979460705932286695355, −6.14352382034057490290990421258, −5.69504964064351602144460072156, −4.80591948104383450471270261133, −4.07088398213496117407664420317, −3.55523944292039785373435042107, −2.04454010798360477649298658347, −1.48525473096577973706151091988, −0.16472322014360683173332056853,
1.21109595020964426405112146023, 2.69615424271880053651900226617, 3.03455951428175489244179491749, 3.69869899148043290602319778187, 4.92599643154723222860600478546, 5.66227919834459264111869571897, 6.25279087281403197588905787413, 7.16530675738397937874874698741, 7.44297268923464151392858880437, 8.289451056125251814807567253710