L(s) = 1 | + (1.73 − 1.41i)5-s − 4.24·7-s − 2.44·11-s + 2.44i·13-s + 6.92·17-s + 6.92i·19-s − 6i·23-s + (0.999 − 4.89i)25-s + 2.82i·29-s − 3.46i·31-s + (−7.34 + 6i)35-s − 2.44i·37-s + 7.07i·41-s + 8.48·43-s + 10.9·49-s + ⋯ |
L(s) = 1 | + (0.774 − 0.632i)5-s − 1.60·7-s − 0.738·11-s + 0.679i·13-s + 1.68·17-s + 1.58i·19-s − 1.25i·23-s + (0.199 − 0.979i)25-s + 0.525i·29-s − 0.622i·31-s + (−1.24 + 1.01i)35-s − 0.402i·37-s + 1.10i·41-s + 1.29·43-s + 1.57·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.633490306\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.633490306\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.73 + 1.41i)T \) |
good | 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 - 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 2.44iT - 37T^{2} \) |
| 41 | \( 1 - 7.07iT - 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 73 | \( 1 + 4.89iT - 73T^{2} \) |
| 79 | \( 1 + 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 7.07iT - 89T^{2} \) |
| 97 | \( 1 + 14.6iT - 97T^{2} \) |
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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
−8.848392953890687310682549715409, −8.037428737963514305693930338256, −7.21767855213179035453721300365, −6.10432326040306967786454065339, −5.97042885137183989929299826025, −4.96129904892172380122292135554, −3.89655320602852384143105860528, −3.06226194867756057401765268584, −2.07569507470788890516003616365, −0.75861915184940834965947655636,
0.790762929631313492287406140967, 2.46050063143127897782753690368, 3.04327326930013401793778065992, 3.74085664139760024224437962530, 5.42031726923020994067739624775, 5.54070569999631527895485959543, 6.61167190176294903089713258741, 7.14976284922796430518754251753, 7.916249591279328131591039211472, 9.044238957655121180326551120384