| L(s) = 1 | − 3-s + 2i·5-s + 2i·7-s + 9-s + (−3 − 2i)13-s − 2i·15-s − 2·17-s + 6i·19-s − 2i·21-s − 4·23-s + 25-s − 27-s − 10·29-s − 10i·31-s − 4·35-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894i·5-s + 0.755i·7-s + 0.333·9-s + (−0.832 − 0.554i)13-s − 0.516i·15-s − 0.485·17-s + 1.37i·19-s − 0.436i·21-s − 0.834·23-s + 0.200·25-s − 0.192·27-s − 1.85·29-s − 1.79i·31-s − 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.190879 + 0.630433i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.190879 + 0.630433i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + (3 + 2i)T \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 10iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−11.07896983515421853614331169059, −10.04679325876753148338500550929, −9.545256855768720693417805958347, −8.138502611419076158543521176053, −7.45008827257042761708358754418, −6.26849057728637230426964970405, −5.75199585861677739222087556246, −4.53379657942317414986114047907, −3.22174137832237565687415245338, −2.03528827130178337389285144446,
0.36976518988193321614937555599, 1.97511574538071200350409997528, 3.82723900474355564856962302199, 4.76176203090527311101028243801, 5.46681862923474358567772824212, 6.86709670179462603940985686001, 7.33952654961547107359582725575, 8.705910570820401902980974531783, 9.315588980610137988738780657471, 10.36155304252396106225225446237
