L(s) = 1 | + (2.59 + 1.5i)3-s + (−2.59 + 1.5i)7-s + (3 + 5.19i)9-s + (−1.5 + 2.59i)11-s + (−3.46 + i)13-s + (6.06 − 3.5i)17-s + (0.5 + 0.866i)19-s − 9·21-s + (6.06 + 3.5i)23-s + 9i·27-s + (−2.5 + 4.33i)29-s − 4·31-s + (−7.79 + 4.5i)33-s + (−2.59 − 1.5i)37-s + (−10.5 − 2.59i)39-s + ⋯ |
L(s) = 1 | + (1.49 + 0.866i)3-s + (−0.981 + 0.566i)7-s + (1 + 1.73i)9-s + (−0.452 + 0.783i)11-s + (−0.960 + 0.277i)13-s + (1.47 − 0.848i)17-s + (0.114 + 0.198i)19-s − 1.96·21-s + (1.26 + 0.729i)23-s + 1.73i·27-s + (−0.464 + 0.804i)29-s − 0.718·31-s + (−1.35 + 0.783i)33-s + (−0.427 − 0.246i)37-s + (−1.68 − 0.416i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.217848770\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.217848770\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (3.46 - i)T \) |
good | 3 | \( 1 + (-2.59 - 1.5i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (2.59 - 1.5i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-6.06 + 3.5i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.06 - 3.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.5 - 4.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (2.59 + 1.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.5 - 6.06i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (7.79 - 4.5i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + (-2.5 - 4.33i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-11.2 - 6.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.5 - 2.59i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.52 + 5.5i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.839959566340699410845550903493, −9.280805446374085115734139952406, −8.499147717110062427808695341904, −7.49813233681921234583556075093, −7.01543148840224216825071019070, −5.38437232919076095023508226281, −4.82997563268097608512859044859, −3.43849065094132470166468523251, −3.07715631262917029914693650383, −2.01070486842871243403993685576,
0.74415245683020832165539587452, 2.19566575403911816895561774177, 3.23359060522149954992502521669, 3.60297079475479156408754319653, 5.20768396828488599641568330479, 6.37098438775061410126775251949, 7.15039373005845966717277190854, 7.77974349741270454721636439108, 8.449886275596824471331505137575, 9.302767612793782828129742105811