L(s) = 1 | + (1.41 − 1.41i)4-s + (−2.77 − 1.14i)9-s + (2.75 − 0.548i)11-s + (1.77 + 1.77i)13-s − 4.00i·16-s + (3.98 − 1.05i)17-s + (−0.988 − 4.96i)23-s + (1.91 − 4.61i)25-s + (−0.443 − 0.0882i)31-s + (−5.54 + 2.29i)36-s + (−7.90 − 5.28i)41-s + (6.05 + 2.50i)43-s + (3.12 − 4.67i)44-s + (−0.935 − 0.935i)47-s + (2.67 + 6.46i)49-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)4-s + (−0.923 − 0.382i)9-s + (0.831 − 0.165i)11-s + (0.493 + 0.493i)13-s − 1.00i·16-s + (0.966 − 0.255i)17-s + (−0.206 − 1.03i)23-s + (0.382 − 0.923i)25-s + (−0.0796 − 0.0158i)31-s + (−0.923 + 0.382i)36-s + (−1.23 − 0.825i)41-s + (0.923 + 0.382i)43-s + (0.470 − 0.704i)44-s + (−0.136 − 0.136i)47-s + (0.382 + 0.923i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.51830 - 0.888635i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.51830 - 0.888635i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-3.98 + 1.05i)T \) |
| 43 | \( 1 + (-6.05 - 2.50i)T \) |
good | 2 | \( 1 + (-1.41 + 1.41i)T^{2} \) |
| 3 | \( 1 + (2.77 + 1.14i)T^{2} \) |
| 5 | \( 1 + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (-2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-2.75 + 0.548i)T + (10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (-1.77 - 1.77i)T + 13iT^{2} \) |
| 19 | \( 1 + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (0.988 + 4.96i)T + (-21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 + (0.443 + 0.0882i)T + (28.6 + 11.8i)T^{2} \) |
| 37 | \( 1 + (34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (7.90 + 5.28i)T + (15.6 + 37.8i)T^{2} \) |
| 47 | \( 1 + (0.935 + 0.935i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.00 - 0.414i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (5.80 - 14.0i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + 16.3iT - 67T^{2} \) |
| 71 | \( 1 + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-14.0 + 2.79i)T + (72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (-2.50 - 6.05i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + 89iT^{2} \) |
| 97 | \( 1 + (2.67 + 4.00i)T + (-37.1 + 89.6i)T^{2} \) |
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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
−10.39091736782749225440929869227, −9.353266359579751199295770700373, −8.687109857957817285239969629355, −7.56485929286055877064538131204, −6.41058789045243355006136061590, −6.09561625428432192347132114399, −4.91780511139432617750338077035, −3.57880865783932753836423465068, −2.42621406795068358130534023001, −0.977055366957075231561650685697,
1.64158720805134176462891535082, 3.06454655998421704355070936476, 3.74192659700056524068674739800, 5.29441855617555850804954481638, 6.13434074997621124870335806169, 7.12840364266308489868556510799, 7.966736497581805911881530317475, 8.636484669217394581073020271609, 9.662299133182291221919492817411, 10.73909340529681271490567570888