L(s) = 1 | + (0.988 − 0.149i)7-s + (0.880 − 0.702i)13-s + (−0.258 − 0.149i)19-s + (−0.988 − 0.149i)25-s + (−0.510 + 0.294i)31-s + (1.40 + 0.432i)37-s + (0.658 + 0.317i)43-s + (0.955 − 0.294i)49-s + (0.574 − 1.86i)61-s + (−0.0747 − 0.129i)67-s + (−0.202 + 1.34i)73-s + (−0.955 + 1.65i)79-s + (0.766 − 0.825i)91-s − 1.56i·97-s + (−1.85 − 0.139i)103-s + ⋯ |
L(s) = 1 | + (0.988 − 0.149i)7-s + (0.880 − 0.702i)13-s + (−0.258 − 0.149i)19-s + (−0.988 − 0.149i)25-s + (−0.510 + 0.294i)31-s + (1.40 + 0.432i)37-s + (0.658 + 0.317i)43-s + (0.955 − 0.294i)49-s + (0.574 − 1.86i)61-s + (−0.0747 − 0.129i)67-s + (−0.202 + 1.34i)73-s + (−0.955 + 1.65i)79-s + (0.766 − 0.825i)91-s − 1.56i·97-s + (−1.85 − 0.139i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.293317238\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.293317238\) |
\(L(1)\) |
|
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 | \( 1 + (-0.988 + 0.149i)T \) |
good | 5 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 11 | \( 1 + (-0.733 - 0.680i)T^{2} \) |
| 13 | \( 1 + (-0.880 + 0.702i)T + (0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 19 | \( 1 + (0.258 + 0.149i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.0747 + 0.997i)T^{2} \) |
| 29 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (0.510 - 0.294i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.40 - 0.432i)T + (0.826 + 0.563i)T^{2} \) |
| 41 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 43 | \( 1 + (-0.658 - 0.317i)T + (0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 53 | \( 1 + (0.826 - 0.563i)T^{2} \) |
| 59 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 61 | \( 1 + (-0.574 + 1.86i)T + (-0.826 - 0.563i)T^{2} \) |
| 67 | \( 1 + (0.0747 + 0.129i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (0.202 - 1.34i)T + (-0.955 - 0.294i)T^{2} \) |
| 79 | \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 97 | \( 1 + 1.56iT - T^{2} \) |
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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
−9.472155240863141165109132029862, −8.449941961159471632655182516293, −8.045063231168474062046732171404, −7.19339392090971013915631400051, −6.14968883095960434114865146671, −5.44146307880557167735088509523, −4.48262829426868045580289846506, −3.65569597363906581748545626953, −2.42610432357003008650800456132, −1.22579082936470773715311315130,
1.43144871964847281885777039486, 2.42304040440377002549256069704, 3.83741175307538339416856120707, 4.45597501185927854456364169275, 5.57029102228211600360147402825, 6.18519634096975817147301112899, 7.29322980979943306206455952451, 7.944434079479631561771895277033, 8.759995930752658773985104384435, 9.355121163035715106159764066293