L(s) = 1 | + (0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + (−0.569 − 2.16i)5-s + 1.00i·6-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (1.12 − 1.93i)10-s − 2.01·11-s + (−0.707 + 0.707i)12-s + (3.44 + 3.44i)13-s + (1.12 − 1.93i)15-s − 1.00·16-s + (3.40 − 3.40i)17-s + (−0.707 + 0.707i)18-s + 6.72·19-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (−0.254 − 0.966i)5-s + 0.408i·6-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (0.356 − 0.610i)10-s − 0.607·11-s + (−0.204 + 0.204i)12-s + (0.954 + 0.954i)13-s + (0.290 − 0.498i)15-s − 0.250·16-s + (0.824 − 0.824i)17-s + (−0.166 + 0.166i)18-s + 1.54·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.560819979\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.560819979\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.569 + 2.16i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2.01T + 11T^{2} \) |
| 13 | \( 1 + (-3.44 - 3.44i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.40 + 3.40i)T - 17iT^{2} \) |
| 19 | \( 1 - 6.72T + 19T^{2} \) |
| 23 | \( 1 + (-4.65 + 4.65i)T - 23iT^{2} \) |
| 29 | \( 1 - 1.60iT - 29T^{2} \) |
| 31 | \( 1 - 6.25iT - 31T^{2} \) |
| 37 | \( 1 + (-3.63 - 3.63i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.94iT - 41T^{2} \) |
| 43 | \( 1 + (4.68 - 4.68i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.07 - 1.07i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.42 + 1.42i)T - 53iT^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 9.00iT - 61T^{2} \) |
| 67 | \( 1 + (1.19 + 1.19i)T + 67iT^{2} \) |
| 71 | \( 1 + 3.49T + 71T^{2} \) |
| 73 | \( 1 + (-5.97 - 5.97i)T + 73iT^{2} \) |
| 79 | \( 1 + 13.9iT - 79T^{2} \) |
| 83 | \( 1 + (8.59 + 8.59i)T + 83iT^{2} \) |
| 89 | \( 1 - 2.40T + 89T^{2} \) |
| 97 | \( 1 + (-13.1 + 13.1i)T - 97iT^{2} \) |
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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
−9.382664515039381557342395399866, −8.775926910776084425487752076013, −8.051611305300310347214899715820, −7.30514602714017926112875974396, −6.31189675464645731316448288335, −5.06509935523403313776685365120, −4.91394682550245085418666556702, −3.69924421119195664785603844940, −2.91136328930635370463895934832, −1.21595988038675845777326487127,
1.04310209863073920489897138161, 2.43705960127092727274448564975, 3.32227081024154629523415582018, 3.80856244597964739009220702146, 5.44811850078426686570098063400, 5.86748239533610004411432674689, 7.08734424030142232994662221191, 7.67970823009817018051282933701, 8.448126967310616644609545433165, 9.637941583523398149233672165985