| L(s) = 1 | + i·3-s + (2.14 + 0.619i)5-s + 1.66i·7-s − 9-s − 5.96·11-s − 3.02i·13-s + (−0.619 + 2.14i)15-s + 6.90i·17-s − 6.93·19-s − 1.66·21-s − i·23-s + (4.23 + 2.66i)25-s − i·27-s − 7.66·29-s − 5.56·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.960 + 0.276i)5-s + 0.627i·7-s − 0.333·9-s − 1.79·11-s − 0.840i·13-s + (−0.159 + 0.554i)15-s + 1.67i·17-s − 1.58·19-s − 0.362·21-s − 0.208i·23-s + (0.846 + 0.532i)25-s − 0.192i·27-s − 1.42·29-s − 0.999·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9152640274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9152640274\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2.14 - 0.619i)T \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 - 1.66iT - 7T^{2} \) |
| 11 | \( 1 + 5.96T + 11T^{2} \) |
| 13 | \( 1 + 3.02iT - 13T^{2} \) |
| 17 | \( 1 - 6.90iT - 17T^{2} \) |
| 19 | \( 1 + 6.93T + 19T^{2} \) |
| 29 | \( 1 + 7.66T + 29T^{2} \) |
| 31 | \( 1 + 5.56T + 31T^{2} \) |
| 37 | \( 1 - 6.17iT - 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 - 3.84iT - 43T^{2} \) |
| 47 | \( 1 + 7.29iT - 47T^{2} \) |
| 53 | \( 1 - 11.8iT - 53T^{2} \) |
| 59 | \( 1 - 5.53T + 59T^{2} \) |
| 61 | \( 1 - 1.81T + 61T^{2} \) |
| 67 | \( 1 + 9.32iT - 67T^{2} \) |
| 71 | \( 1 - 9.14T + 71T^{2} \) |
| 73 | \( 1 + 8.35iT - 73T^{2} \) |
| 79 | \( 1 + 3.89T + 79T^{2} \) |
| 83 | \( 1 - 12.1iT - 83T^{2} \) |
| 89 | \( 1 + 8.30T + 89T^{2} \) |
| 97 | \( 1 - 10.7iT - 97T^{2} \) |
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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
−10.11836457958487289634382679695, −9.176028471033140655380858216421, −8.402383876917173620840974356601, −7.73238530992796420510607550976, −6.38777234046870198276955020100, −5.68329218607572226943809219105, −5.19143378507245059136084614535, −3.92650421513254440933400541022, −2.74437102321910877405386505485, −2.02828400274943869490914235310,
0.32980977911818927437111384469, 1.96580184058460448280733096212, 2.62119692228413765274365136842, 4.14286246966081962505852601518, 5.21059992447814820351081163172, 5.77821616375797912530408346050, 6.97004609408072133291162566403, 7.38676384831847201208599466055, 8.422479070341500602141729414480, 9.243736420348307716523737328539
