L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (2.16 + 0.561i)5-s + (2.06 + 1.65i)7-s + (−0.707 − 0.707i)8-s + (0.0180 + 2.23i)10-s + (0.565 − 0.326i)11-s + (0.771 − 0.771i)13-s + (−1.06 + 2.42i)14-s + (0.500 − 0.866i)16-s + (−2.06 − 0.554i)17-s + (4.34 + 2.50i)19-s + (−2.15 + 0.596i)20-s + (0.461 + 0.461i)22-s + (0.905 − 0.242i)23-s + ⋯ |
L(s) = 1 | + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (0.967 + 0.251i)5-s + (0.779 + 0.626i)7-s + (−0.249 − 0.249i)8-s + (0.00570 + 0.707i)10-s + (0.170 − 0.0984i)11-s + (0.214 − 0.214i)13-s + (−0.285 + 0.647i)14-s + (0.125 − 0.216i)16-s + (−0.501 − 0.134i)17-s + (0.996 + 0.575i)19-s + (−0.481 + 0.133i)20-s + (0.0984 + 0.0984i)22-s + (0.188 − 0.0505i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.143 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.143 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49466 + 1.29365i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49466 + 1.29365i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.16 - 0.561i)T \) |
| 7 | \( 1 + (-2.06 - 1.65i)T \) |
good | 11 | \( 1 + (-0.565 + 0.326i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.771 + 0.771i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.06 + 0.554i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-4.34 - 2.50i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.905 + 0.242i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 9.79T + 29T^{2} \) |
| 31 | \( 1 + (0.897 + 1.55i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.26 + 1.67i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 7.89iT - 41T^{2} \) |
| 43 | \( 1 + (0.0657 - 0.0657i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.854 + 3.18i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (1.91 - 7.13i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (3.39 + 5.87i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.33 - 7.51i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.66 + 6.21i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 7.68iT - 71T^{2} \) |
| 73 | \( 1 + (9.62 + 2.58i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-10.3 - 5.96i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.838 + 0.838i)T + 83iT^{2} \) |
| 89 | \( 1 + (-8.65 + 14.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.59 + 8.59i)T + 97iT^{2} \) |
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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
−10.84838812126509619718713198403, −9.618714111709893543233653481553, −9.120771368202165261766492650357, −8.067176732858990449595090051900, −7.25261984642158468457637763427, −6.05489352569633320424801922397, −5.58015690991045548038981195954, −4.54639826330502954862172280500, −3.08551413009730621947070743926, −1.70856937670880723893518262668,
1.21161393274559243892031955572, 2.28047970178620591640436526799, 3.74342192049626539959849278730, 4.81450641721845201925314201840, 5.58579903478576242939603594223, 6.77762232160236759774385056597, 7.82743017429855926203279221247, 9.016497863870744627159369289116, 9.515381761716386825880058977076, 10.55456153259392593686286451084