L(s) = 1 | + (0.330 − 0.943i)2-s + (−0.362 − 0.577i)3-s + (−0.781 − 0.623i)4-s + (1.73 − 1.41i)5-s + (−0.664 + 0.151i)6-s + (−2.08 − 1.62i)7-s + (−0.846 + 0.532i)8-s + (1.10 − 2.28i)9-s + (−0.764 − 2.10i)10-s + (−0.921 + 0.443i)11-s + (−0.0763 + 0.677i)12-s + (−0.713 + 2.03i)13-s + (−2.22 + 1.43i)14-s + (−1.44 − 0.485i)15-s + (0.222 + 0.974i)16-s + (0.178 − 1.58i)17-s + ⋯ |
L(s) = 1 | + (0.233 − 0.667i)2-s + (−0.209 − 0.333i)3-s + (−0.390 − 0.311i)4-s + (0.773 − 0.633i)5-s + (−0.271 + 0.0619i)6-s + (−0.788 − 0.614i)7-s + (−0.299 + 0.188i)8-s + (0.366 − 0.761i)9-s + (−0.241 − 0.664i)10-s + (−0.277 + 0.133i)11-s + (−0.0220 + 0.195i)12-s + (−0.197 + 0.565i)13-s + (−0.594 + 0.382i)14-s + (−0.373 − 0.125i)15-s + (0.0556 + 0.243i)16-s + (0.0432 − 0.383i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.278174 - 1.28383i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.278174 - 1.28383i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.330 + 0.943i)T \) |
| 5 | \( 1 + (-1.73 + 1.41i)T \) |
| 7 | \( 1 + (2.08 + 1.62i)T \) |
good | 3 | \( 1 + (0.362 + 0.577i)T + (-1.30 + 2.70i)T^{2} \) |
| 11 | \( 1 + (0.921 - 0.443i)T + (6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (0.713 - 2.03i)T + (-10.1 - 8.10i)T^{2} \) |
| 17 | \( 1 + (-0.178 + 1.58i)T + (-16.5 - 3.78i)T^{2} \) |
| 19 | \( 1 - 0.760T + 19T^{2} \) |
| 23 | \( 1 + (3.53 - 0.398i)T + (22.4 - 5.11i)T^{2} \) |
| 29 | \( 1 + (-0.279 + 0.222i)T + (6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + 6.31iT - 31T^{2} \) |
| 37 | \( 1 + (-2.16 - 0.244i)T + (36.0 + 8.23i)T^{2} \) |
| 41 | \( 1 + (-2.96 - 0.677i)T + (36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (0.499 - 0.795i)T + (-18.6 - 38.7i)T^{2} \) |
| 47 | \( 1 + (-1.21 - 0.423i)T + (36.7 + 29.3i)T^{2} \) |
| 53 | \( 1 + (6.84 - 0.771i)T + (51.6 - 11.7i)T^{2} \) |
| 59 | \( 1 + (-1.52 - 6.69i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (-10.6 + 8.53i)T + (13.5 - 59.4i)T^{2} \) |
| 67 | \( 1 + (4.39 - 4.39i)T - 67iT^{2} \) |
| 71 | \( 1 + (-10.4 + 13.1i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-13.6 + 4.77i)T + (57.0 - 45.5i)T^{2} \) |
| 79 | \( 1 - 2.99iT - 79T^{2} \) |
| 83 | \( 1 + (-2.95 + 1.03i)T + (64.8 - 51.7i)T^{2} \) |
| 89 | \( 1 + (-10.6 - 5.14i)T + (55.4 + 69.5i)T^{2} \) |
| 97 | \( 1 + (-4.64 - 4.64i)T + 97iT^{2} \) |
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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
−10.49148098327349810456736231663, −9.557022729108196448460827945013, −9.368859483727875022167118840944, −7.84627263358286330443929164970, −6.65568793936678301652438500367, −5.92607181831084299810210694093, −4.69063699574623976874537371798, −3.67290924799009907795330906278, −2.18833920664267517923171005572, −0.75547458424188438274164147745,
2.35305412522409374424216065564, 3.54387684036920776798071537125, 5.03754467369599792167198191884, 5.75865248085453978607198304176, 6.58357529606097040822110259142, 7.58266167167237112636204435454, 8.618549688104432216304268284511, 9.750665038531930520251335376844, 10.22440457727034440934343422708, 11.18307062439287800032659966252