L(s) = 1 | − 1.41i·2-s + (−0.275 + 0.158i)3-s − 2.00·4-s + (−0.5 + 0.866i)5-s + (0.224 + 0.389i)6-s + (−2.5 − 0.866i)7-s + 2.82i·8-s + (−1.44 + 2.51i)9-s + (1.22 + 0.707i)10-s + (−2.44 − 4.24i)11-s + (0.550 − 0.317i)12-s − 4.44·13-s + (−1.22 + 3.53i)14-s − 0.317i·15-s + 4.00·16-s + (−4.22 + 2.43i)17-s + ⋯ |
L(s) = 1 | − 0.999i·2-s + (−0.158 + 0.0917i)3-s − 1.00·4-s + (−0.223 + 0.387i)5-s + (0.0917 + 0.158i)6-s + (−0.944 − 0.327i)7-s + 1.00i·8-s + (−0.483 + 0.836i)9-s + (0.387 + 0.223i)10-s + (−0.738 − 1.27i)11-s + (0.158 − 0.0917i)12-s − 1.23·13-s + (−0.327 + 0.944i)14-s − 0.0820i·15-s + 1.00·16-s + (−1.02 + 0.591i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 3 | \( 1 + (0.275 - 0.158i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (2.44 + 4.24i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.44T + 13T^{2} \) |
| 17 | \( 1 + (4.22 - 2.43i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.67 - 2.12i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.94 - 2.28i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.24iT - 29T^{2} \) |
| 31 | \( 1 + (0.775 + 1.34i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3 + 1.73i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8.02iT - 41T^{2} \) |
| 43 | \( 1 + 9.44T + 43T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.77 - 1.02i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.12 + 1.80i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.174 + 0.301i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.17 + 10.6i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.41iT - 71T^{2} \) |
| 73 | \( 1 + (9.67 - 5.58i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.67 + 3.85i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.87iT - 83T^{2} \) |
| 89 | \( 1 + (-9.39 - 5.42i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 11.9iT - 97T^{2} \) |
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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
−11.18459593449970809714227803554, −10.43597722252966114039167660996, −9.680328219812060745610158771733, −8.482326698885140156043849878094, −7.53293967222389299818625715395, −5.98519094135951549798139863897, −4.90601835855517239878397692751, −3.46970536384755394067781227721, −2.53422944431611399476909669148, 0,
3.00268188933097995799999720595, 4.66926394236766159527419242616, 5.42183200585615445833258153681, 6.90435500159865840459380640733, 7.19732660168812778337160342315, 8.791461889856836695613114174551, 9.341104589415946978843128269344, 10.24814643986683716249685327374, 11.92390969901690724386654198877