L(s) = 1 | − 1.77i·5-s + (−2.39 − 1.12i)7-s − 3.85·11-s + 6.59·13-s + 4.56·17-s + 1.05·19-s − 0.0946i·23-s + 1.83·25-s + 4.31·29-s − 4.07i·31-s + (−2.00 + 4.25i)35-s + 4.65i·37-s − 11.5·41-s − 6.28i·43-s + 6.12·47-s + ⋯ |
L(s) = 1 | − 0.794i·5-s + (−0.904 − 0.426i)7-s − 1.16·11-s + 1.82·13-s + 1.10·17-s + 0.241·19-s − 0.0197i·23-s + 0.367·25-s + 0.801·29-s − 0.732i·31-s + (−0.339 + 0.718i)35-s + 0.764i·37-s − 1.80·41-s − 0.958i·43-s + 0.894·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0873 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0873 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.554154117\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.554154117\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.39 + 1.12i)T \) |
good | 5 | \( 1 + 1.77iT - 5T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 - 6.59T + 13T^{2} \) |
| 17 | \( 1 - 4.56T + 17T^{2} \) |
| 19 | \( 1 - 1.05T + 19T^{2} \) |
| 23 | \( 1 + 0.0946iT - 23T^{2} \) |
| 29 | \( 1 - 4.31T + 29T^{2} \) |
| 31 | \( 1 + 4.07iT - 31T^{2} \) |
| 37 | \( 1 - 4.65iT - 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 + 6.28iT - 43T^{2} \) |
| 47 | \( 1 - 6.12T + 47T^{2} \) |
| 53 | \( 1 - 2.44T + 53T^{2} \) |
| 59 | \( 1 - 13.1iT - 59T^{2} \) |
| 61 | \( 1 + 4.58T + 61T^{2} \) |
| 67 | \( 1 + 4.83iT - 67T^{2} \) |
| 71 | \( 1 + 11.5iT - 71T^{2} \) |
| 73 | \( 1 + 2.25iT - 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 10.6iT - 83T^{2} \) |
| 89 | \( 1 + 8.61T + 89T^{2} \) |
| 97 | \( 1 + 14.2iT - 97T^{2} \) |
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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
−8.323595558971637379617526368745, −7.62507143589305550834575261055, −6.74126682192116868757671021247, −5.94415695941834423120118467156, −5.36752348834455996857467734787, −4.44691113092195803073891269170, −3.53367237306534964959618781854, −2.92473164958543490875760683393, −1.45701000356085437698248743649, −0.53243325523167281652515848460,
1.08273409659612147377699226287, 2.48817210687334131332660780640, 3.22938968768168698364336580954, 3.71203472798945521237076828145, 5.10631783721104276524636925239, 5.73250859616373978431682182052, 6.47911152337110122294175505911, 7.00823751827135321742312250240, 8.032052195762936262171707738659, 8.495447737702253901689026435749