L(s) = 1 | + (1.5 − 0.866i)3-s + (−1 + 1.73i)5-s + (−0.5 − 0.866i)7-s + (1.5 − 2.59i)9-s + (1.5 + 2.59i)11-s + (3 − 5.19i)13-s + 3.46i·15-s + 7·17-s + 19-s + (−1.5 − 0.866i)21-s + (2 − 3.46i)23-s + (0.500 + 0.866i)25-s − 5.19i·27-s + (2 + 3.46i)29-s + (−5 + 8.66i)31-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)3-s + (−0.447 + 0.774i)5-s + (−0.188 − 0.327i)7-s + (0.5 − 0.866i)9-s + (0.452 + 0.783i)11-s + (0.832 − 1.44i)13-s + 0.894i·15-s + 1.69·17-s + 0.229·19-s + (−0.327 − 0.188i)21-s + (0.417 − 0.722i)23-s + (0.100 + 0.173i)25-s − 0.999i·27-s + (0.371 + 0.643i)29-s + (−0.898 + 1.55i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.85315 - 0.326761i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.85315 - 0.326761i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 7T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + (3.5 - 6.06i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.5 - 7.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 13T + 73T^{2} \) |
| 79 | \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.57987172856091387269307319921, −10.16651189806828001197810344513, −8.937015850406098283432099225085, −8.073881502440707503240949903814, −7.25588369781986287338199477485, −6.64587556181450672197171308630, −5.22838268089954473354893221682, −3.48335615216255018675711907617, −3.21091617412705841993752115561, −1.35037306117712095951213360578,
1.52404278828785813517052309157, 3.27389694804162546342411492395, 4.02125640619201736529774437366, 5.14941804038973511286447557072, 6.31374737523508490450492597426, 7.68998471435857884591269073388, 8.392145895600724234114854847862, 9.208792837909785447575518841160, 9.711745103301385996854714350953, 11.05650201104948513639400187977