| L(s) = 1 | + (−0.959 + 0.281i)2-s + (0.396 + 2.76i)3-s + (0.841 − 0.540i)4-s + (−0.415 + 0.909i)5-s + (−1.15 − 2.53i)6-s + (−2.99 + 0.880i)7-s + (−0.654 + 0.755i)8-s + (−4.58 + 1.34i)9-s + (0.142 − 0.989i)10-s + (0.459 − 1.00i)11-s + (1.82 + 2.10i)12-s + (0.511 + 0.590i)13-s + (2.62 − 1.68i)14-s + (−2.67 − 0.785i)15-s + (0.415 − 0.909i)16-s + (−1.75 − 1.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (0.229 + 1.59i)3-s + (0.420 − 0.270i)4-s + (−0.185 + 0.406i)5-s + (−0.473 − 1.03i)6-s + (−1.13 + 0.332i)7-s + (−0.231 + 0.267i)8-s + (−1.52 + 0.448i)9-s + (0.0450 − 0.313i)10-s + (0.138 − 0.303i)11-s + (0.527 + 0.608i)12-s + (0.141 + 0.163i)13-s + (0.702 − 0.451i)14-s + (−0.691 − 0.202i)15-s + (0.103 − 0.227i)16-s + (−0.424 − 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.190572 - 0.318696i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.190572 - 0.318696i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 5 | \( 1 + (0.415 - 0.909i)T \) |
| 67 | \( 1 + (5.35 - 6.19i)T \) |
| good | 3 | \( 1 + (-0.396 - 2.76i)T + (-2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (2.99 - 0.880i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-0.459 + 1.00i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.511 - 0.590i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.75 + 1.12i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (3.98 + 1.17i)T + (15.9 + 10.2i)T^{2} \) |
| 23 | \( 1 + (-0.193 - 1.34i)T + (-22.0 + 6.47i)T^{2} \) |
| 29 | \( 1 + 5.42T + 29T^{2} \) |
| 31 | \( 1 + (-2.25 + 2.59i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 - 3.28T + 37T^{2} \) |
| 41 | \( 1 + (-1.34 - 0.864i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (0.880 + 0.565i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (-1.69 - 11.7i)T + (-45.0 + 13.2i)T^{2} \) |
| 53 | \( 1 + (-2.68 + 1.72i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (2.75 - 3.17i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (5.12 + 11.2i)T + (-39.9 + 46.1i)T^{2} \) |
| 71 | \( 1 + (-7.51 + 4.83i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (4.41 + 9.66i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (3.43 + 3.95i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (3.23 - 7.09i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (0.784 - 5.45i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−10.88188798699265248929786438403, −10.03521465152377466938188757829, −9.340190736925903852884824287742, −8.934638329916148802784522526869, −7.81028914086011308845960916317, −6.57074093126435361792079331169, −5.84108182491586047310558308854, −4.53301427402758593653065471044, −3.53674653363917844900972756141, −2.60982477571466461919023124064,
0.23096096718435029156381887581, 1.59759388461227810593454087102, 2.73427195388099926624450426701, 4.00977687210639650912726947531, 5.86200668441838071596165787416, 6.70118235068816000768454416649, 7.25300550350916377779623034736, 8.225255237735709358634948787085, 8.842829498641608677978419617934, 9.829281240825421603963980439077
