| L(s) = 1 | + (2.87 + 0.863i)3-s + 4.52i·5-s + (−1.50 + 2.60i)7-s + (7.50 + 4.96i)9-s + (12.2 + 7.05i)11-s + (−6.36 + 11.0i)13-s + (−3.90 + 13.0i)15-s + (−5.51 − 3.18i)17-s + (−14.3 − 12.4i)19-s + (−6.56 + 6.17i)21-s + (−13.4 − 7.75i)23-s + 4.52·25-s + (17.2 + 20.7i)27-s − 5.02i·29-s + (−1.24 − 2.16i)31-s + ⋯ |
| L(s) = 1 | + (0.957 + 0.287i)3-s + 0.905i·5-s + (−0.214 + 0.371i)7-s + (0.834 + 0.551i)9-s + (1.11 + 0.641i)11-s + (−0.489 + 0.847i)13-s + (−0.260 + 0.866i)15-s + (−0.324 − 0.187i)17-s + (−0.755 − 0.655i)19-s + (−0.312 + 0.294i)21-s + (−0.583 − 0.337i)23-s + 0.180·25-s + (0.640 + 0.768i)27-s − 0.173i·29-s + (−0.0402 − 0.0697i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.366530852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.366530852\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.87 - 0.863i)T \) |
| 19 | \( 1 + (14.3 + 12.4i)T \) |
| good | 5 | \( 1 - 4.52iT - 25T^{2} \) |
| 7 | \( 1 + (1.50 - 2.60i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-12.2 - 7.05i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (6.36 - 11.0i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (5.51 + 3.18i)T + (144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (13.4 + 7.75i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 5.02iT - 841T^{2} \) |
| 31 | \( 1 + (1.24 + 2.16i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 47.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 38.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-34.7 - 60.2i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + 27.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (31.0 - 17.9i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + 53.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 64.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-16.9 + 29.2i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (0.687 + 0.397i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-25.7 + 44.6i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-32.8 - 56.8i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-23.4 - 13.5i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-22.7 + 13.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-73.1 - 126. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.40173096423949179434084188968, −9.476321209847884542321195299702, −9.061408545277605357888669113058, −7.967021386062662758291738072943, −6.88460623282816669771291972186, −6.50567012237657629440135819809, −4.75584235729946578706250367407, −3.95339574368814785019105064500, −2.78658473623936005159800796596, −1.91944471764343677078270662618,
0.74960210994503130582744954384, 1.98298449234383902049308237308, 3.47237306245326576626272410859, 4.17341134001274453262239075007, 5.48879313770425887656314226687, 6.60787162193466803605139346364, 7.51011957280453637715108415159, 8.552561416421793848344523858155, 8.856372785211219433547324383895, 9.888384874474667201705384647118
