L(s) = 1 | + 0.284i·3-s + 2.19·7-s + 8.91·9-s − 2.14·11-s + 5.91i·13-s − 21.3·17-s + (−12.8 − 13.9i)19-s + 0.624i·21-s − 9.06·23-s + 5.09i·27-s + 33.2i·29-s + 44.1i·31-s − 0.610i·33-s + 50.0i·37-s − 1.68·39-s + ⋯ |
L(s) = 1 | + 0.0947i·3-s + 0.313·7-s + 0.991·9-s − 0.195·11-s + 0.454i·13-s − 1.25·17-s + (−0.676 − 0.736i)19-s + 0.0297i·21-s − 0.393·23-s + 0.188i·27-s + 1.14i·29-s + 1.42i·31-s − 0.0185i·33-s + 1.35i·37-s − 0.0430·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.008634241\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.008634241\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (12.8 + 13.9i)T \) |
good | 3 | \( 1 - 0.284iT - 9T^{2} \) |
| 7 | \( 1 - 2.19T + 49T^{2} \) |
| 11 | \( 1 + 2.14T + 121T^{2} \) |
| 13 | \( 1 - 5.91iT - 169T^{2} \) |
| 17 | \( 1 + 21.3T + 289T^{2} \) |
| 23 | \( 1 + 9.06T + 529T^{2} \) |
| 29 | \( 1 - 33.2iT - 841T^{2} \) |
| 31 | \( 1 - 44.1iT - 961T^{2} \) |
| 37 | \( 1 - 50.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 61.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 5.39T + 1.84e3T^{2} \) |
| 47 | \( 1 - 9.02T + 2.20e3T^{2} \) |
| 53 | \( 1 - 62.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 37.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 58.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 121. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 103. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 120.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 57.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 47.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 68.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 22.7iT - 9.40e3T^{2} \) |
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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
−9.070416610040393133530125644057, −8.784468460006723676834923407862, −7.67787424880358631539041447826, −6.89471312650007999083209490290, −6.37164166667431551475521353415, −4.99242053552971204413427113920, −4.57169357006327340440302811172, −3.56188788077416031382301508919, −2.31611160309766750436127003430, −1.36924600490374062555920837679,
0.24948198638588379293266168500, 1.68285185701027762520395097908, 2.53141738231530064926006527018, 4.00311866265436912224640962464, 4.42022952789229960298520014633, 5.59836956951634221958914385054, 6.36824323454150064079798187808, 7.21566499376529029320580801208, 8.000037715059830809780232998071, 8.578410073178455456484766734184