L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + (−1.49 + 1.66i)5-s − 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (2.23 − 0.122i)10-s − 0.520·11-s + (−0.707 + 0.707i)12-s + (−2.39 − 2.39i)13-s + (−2.23 + 0.122i)15-s − 1.00·16-s + (0.110 − 0.110i)17-s + (0.707 − 0.707i)18-s − 6.73·19-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (−0.667 + 0.744i)5-s − 0.408i·6-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (0.706 − 0.0386i)10-s − 0.156·11-s + (−0.204 + 0.204i)12-s + (−0.663 − 0.663i)13-s + (−0.576 + 0.0315i)15-s − 0.250·16-s + (0.0267 − 0.0267i)17-s + (0.166 − 0.166i)18-s − 1.54·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4197167263\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4197167263\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (1.49 - 1.66i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 0.520T + 11T^{2} \) |
| 13 | \( 1 + (2.39 + 2.39i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.110 + 0.110i)T - 17iT^{2} \) |
| 19 | \( 1 + 6.73T + 19T^{2} \) |
| 23 | \( 1 + (-0.802 + 0.802i)T - 23iT^{2} \) |
| 29 | \( 1 + 1.20iT - 29T^{2} \) |
| 31 | \( 1 + 7.18iT - 31T^{2} \) |
| 37 | \( 1 + (-4.41 - 4.41i)T + 37iT^{2} \) |
| 41 | \( 1 + 1.23iT - 41T^{2} \) |
| 43 | \( 1 + (-6.27 + 6.27i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.57 - 7.57i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.550 + 0.550i)T - 53iT^{2} \) |
| 59 | \( 1 - 8.93T + 59T^{2} \) |
| 61 | \( 1 + 15.4iT - 61T^{2} \) |
| 67 | \( 1 + (10.4 + 10.4i)T + 67iT^{2} \) |
| 71 | \( 1 - 3.05T + 71T^{2} \) |
| 73 | \( 1 + (3.22 + 3.22i)T + 73iT^{2} \) |
| 79 | \( 1 + 2.29iT - 79T^{2} \) |
| 83 | \( 1 + (-3.43 - 3.43i)T + 83iT^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + (9.40 - 9.40i)T - 97iT^{2} \) |
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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
−9.397631840080309395531934706454, −8.289650580984055993457168499375, −7.934797258877830542124558923442, −7.03091764344298912706347164769, −6.07340479572662674579381101323, −4.69585429611778725986130013418, −3.93698110907697947733130878959, −2.94954916672459536501612588700, −2.21257053954957623074654664590, −0.19148171509893983093684336712,
1.31828585875633489871973575531, 2.53707285911341548800854492596, 3.98922482002520817759918123039, 4.75106550006008382126123324374, 5.78959616143872145089996818674, 6.85740081880260091306570659700, 7.38199311579027204038399874352, 8.336145240358795248844763938754, 8.724739019126816432668194545361, 9.494435362295286573520222272956