L(s) = 1 | − i·2-s + (1.54 + 0.780i)3-s − 4-s + (−0.183 + 0.317i)5-s + (0.780 − 1.54i)6-s + i·8-s + (1.78 + 2.41i)9-s + (0.317 + 0.183i)10-s + (−0.579 + 0.334i)11-s + (−1.54 − 0.780i)12-s + (−0.867 + 0.500i)13-s + (−0.531 + 0.347i)15-s + 16-s + (2.49 − 4.32i)17-s + (2.41 − 1.78i)18-s + (5.50 − 3.17i)19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.892 + 0.450i)3-s − 0.5·4-s + (−0.0819 + 0.141i)5-s + (0.318 − 0.631i)6-s + 0.353i·8-s + (0.593 + 0.804i)9-s + (0.100 + 0.0579i)10-s + (−0.174 + 0.100i)11-s + (−0.446 − 0.225i)12-s + (−0.240 + 0.138i)13-s + (−0.137 + 0.0897i)15-s + 0.250·16-s + (0.605 − 1.04i)17-s + (0.569 − 0.419i)18-s + (1.26 − 0.729i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.07192 - 0.130808i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.07192 - 0.130808i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-1.54 - 0.780i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.183 - 0.317i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.579 - 0.334i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.867 - 0.500i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.49 + 4.32i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.50 + 3.17i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.66 - 3.84i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.58 - 0.914i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 37 | \( 1 + (-2.58 - 4.47i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.15 + 3.73i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.24 + 3.89i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 8.32T + 47T^{2} \) |
| 53 | \( 1 + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 8.72T + 59T^{2} \) |
| 61 | \( 1 - 4.95iT - 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 5.49iT - 71T^{2} \) |
| 73 | \( 1 + (-3.52 - 2.03i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 8.35T + 79T^{2} \) |
| 83 | \( 1 + (-8.50 + 14.7i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (5.35 + 9.27i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (14.9 + 8.60i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.00742315735326526191762940362, −9.326168850278394029824411016817, −8.802767309971969291123063213411, −7.58205694754853586528827968746, −7.08168626311138839897826826706, −5.22908961300851847060651397009, −4.76538201052205210245061100614, −3.24182317015346594566907462402, −2.97437293112942028118089563887, −1.38361507077374396532515417267,
1.12076942665013579077635812358, 2.73796071969938767255466687906, 3.76641811351278576992725985801, 4.86607809604842474141058872614, 6.01107386466383809948871414623, 6.81151732371455233951099985092, 7.914914869406396362731832094344, 8.085668536775043996439532479539, 9.201360882125663871480982122809, 9.818297503754599462715682216044