L(s) = 1 | + (−0.707 − 0.707i)3-s + (−2.09 − 0.768i)5-s + (−0.0117 − 0.0117i)7-s + 1.00i·9-s − 1.72i·11-s + (−2.37 − 2.37i)13-s + (0.941 + 2.02i)15-s + (1.66 + 1.66i)17-s − 6.73·19-s + 0.0165i·21-s + (−0.0541 + 4.79i)23-s + (3.81 + 3.22i)25-s + (0.707 − 0.707i)27-s + 0.894i·29-s + 6.90·31-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (−0.939 − 0.343i)5-s + (−0.00443 − 0.00443i)7-s + 0.333i·9-s − 0.520i·11-s + (−0.657 − 0.657i)13-s + (0.243 + 0.523i)15-s + (0.404 + 0.404i)17-s − 1.54·19-s + 0.00361i·21-s + (−0.0112 + 0.999i)23-s + (0.763 + 0.645i)25-s + (0.136 − 0.136i)27-s + 0.166i·29-s + 1.23·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4813333736\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4813333736\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (2.09 + 0.768i)T \) |
| 23 | \( 1 + (0.0541 - 4.79i)T \) |
good | 7 | \( 1 + (0.0117 + 0.0117i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.72iT - 11T^{2} \) |
| 13 | \( 1 + (2.37 + 2.37i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.66 - 1.66i)T + 17iT^{2} \) |
| 19 | \( 1 + 6.73T + 19T^{2} \) |
| 29 | \( 1 - 0.894iT - 29T^{2} \) |
| 31 | \( 1 - 6.90T + 31T^{2} \) |
| 37 | \( 1 + (0.0575 + 0.0575i)T + 37iT^{2} \) |
| 41 | \( 1 - 0.278T + 41T^{2} \) |
| 43 | \( 1 + (-4.31 + 4.31i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.61 - 7.61i)T - 47iT^{2} \) |
| 53 | \( 1 + (9.64 - 9.64i)T - 53iT^{2} \) |
| 59 | \( 1 - 15.1iT - 59T^{2} \) |
| 61 | \( 1 - 5.91iT - 61T^{2} \) |
| 67 | \( 1 + (1.99 + 1.99i)T + 67iT^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + (-7.55 - 7.55i)T + 73iT^{2} \) |
| 79 | \( 1 - 3.91T + 79T^{2} \) |
| 83 | \( 1 + (12.2 - 12.2i)T - 83iT^{2} \) |
| 89 | \( 1 + 6.61T + 89T^{2} \) |
| 97 | \( 1 + (-9.80 - 9.80i)T + 97iT^{2} \) |
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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
−9.830673386550159895497018578346, −8.727069762698494814466271950871, −8.082331222550155275954333593070, −7.46373749349583511356474649507, −6.49784870869176348248731590425, −5.63416073818155848557107594299, −4.71832119755127993248528144553, −3.80545740844616365075343738075, −2.67223910221279363495035617533, −1.12181482796380178133052999031,
0.23619884499033771232223467599, 2.20386527233660480708669406182, 3.38217917980610702315956763662, 4.49712834437715595989820868638, 4.79967919507124330572830754156, 6.35958282302503131184248755150, 6.77276534246747839898335906058, 7.84145257362852156328670422740, 8.491828226020649994753037663433, 9.566307905753201643068059661264