L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.923 + 1.60i)5-s − 0.999i·8-s + (1.60 + 0.923i)10-s − 0.765i·13-s + (−0.5 − 0.866i)16-s + (−0.382 + 0.662i)17-s + 1.84·20-s + (−1.20 + 2.09i)25-s + (−0.382 − 0.662i)26-s + (−0.866 − 0.499i)32-s + 0.765i·34-s + (−0.707 − 1.22i)37-s + (1.60 − 0.923i)40-s + 0.765·41-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.923 + 1.60i)5-s − 0.999i·8-s + (1.60 + 0.923i)10-s − 0.765i·13-s + (−0.5 − 0.866i)16-s + (−0.382 + 0.662i)17-s + 1.84·20-s + (−1.20 + 2.09i)25-s + (−0.382 − 0.662i)26-s + (−0.866 − 0.499i)32-s + 0.765i·34-s + (−0.707 − 1.22i)37-s + (1.60 − 0.923i)40-s + 0.765·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.126541617\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.126541617\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + 0.765iT - T^{2} \) |
| 17 | \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - 0.765T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (1.60 - 0.923i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.60 + 0.923i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + 1.84iT - T^{2} \) |
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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
−9.814495354905709674096300764599, −8.914926578966166700924130948379, −7.48702428855222534343240557190, −6.93637303082961526894799416395, −5.89476955870057440798421982459, −5.76032570795371438607034991162, −4.36401294119883361986638010687, −3.34377010565217061031362112949, −2.66187345004498787139801367263, −1.76214967257629880486954558777,
1.54574740444573382898579389523, 2.58874163696641962113979564878, 4.01464917450545036754445204715, 4.76824448801656204420283114932, 5.32157802493041326130565347426, 6.16004758784143046034202239441, 6.91785698398769212827721805852, 7.968513772560903293682508439818, 8.750531279254604723505556781936, 9.247921968302132748115958352971