L(s) = 1 | + (0.804 − 1.16i)2-s + (0.111 + 0.739i)3-s + (−0.706 − 1.87i)4-s + (2.34 + 0.919i)5-s + (0.949 + 0.465i)6-s + (−2.33 − 1.23i)7-s + (−2.74 − 0.683i)8-s + (2.33 − 0.719i)9-s + (2.95 − 1.98i)10-s + (0.561 − 1.82i)11-s + (1.30 − 0.730i)12-s + (3.11 − 0.711i)13-s + (−3.32 + 1.72i)14-s + (−0.418 + 1.83i)15-s + (−3.00 + 2.64i)16-s + (2.96 − 2.01i)17-s + ⋯ |
L(s) = 1 | + (0.568 − 0.822i)2-s + (0.0643 + 0.426i)3-s + (−0.353 − 0.935i)4-s + (1.04 + 0.411i)5-s + (0.387 + 0.189i)6-s + (−0.883 − 0.468i)7-s + (−0.970 − 0.241i)8-s + (0.777 − 0.239i)9-s + (0.934 − 0.627i)10-s + (0.169 − 0.549i)11-s + (0.376 − 0.210i)12-s + (0.864 − 0.197i)13-s + (−0.887 + 0.460i)14-s + (−0.108 + 0.473i)15-s + (−0.750 + 0.660i)16-s + (0.718 − 0.489i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.68919 - 1.12734i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.68919 - 1.12734i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.804 + 1.16i)T \) |
| 7 | \( 1 + (2.33 + 1.23i)T \) |
good | 3 | \( 1 + (-0.111 - 0.739i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (-2.34 - 0.919i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-0.561 + 1.82i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-3.11 + 0.711i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-2.96 + 2.01i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (0.698 - 0.403i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.36 - 1.61i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (3.29 - 6.84i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-1.00 + 1.74i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (10.4 - 0.782i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (-1.38 - 1.74i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (5.31 + 4.24i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (2.73 - 2.53i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-5.65 - 0.424i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (2.46 - 0.968i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (14.4 - 1.08i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-6.60 - 3.81i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (12.1 - 5.83i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (1.38 + 1.28i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-7.02 - 12.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-12.2 - 2.79i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (5.11 - 1.57i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−10.84697557710739466462535443247, −10.33890174172291474889749227977, −9.630485434490508412152312787146, −8.910186430739771857421099672121, −7.01517151920536424375115982414, −6.16144041914913955246874428707, −5.23435257369103893839625559345, −3.78595215064290748617034608499, −3.10037458412444444905024656476, −1.38456720904869515479936842636,
1.92222209623285002372975511996, 3.51638810937995990042542011987, 4.83363250602431527334662809502, 5.93842208304347060251395775169, 6.51655053049647542204318509712, 7.53331571443729949062157121688, 8.685807295494559447047700635296, 9.453888697764303346692951678562, 10.33395519045264737162195681999, 11.97714103123064710012594780266