L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.295 + 2.62i)7-s + 0.999i·8-s + (0.570 + 0.329i)11-s − 6.13i·13-s + (−1.05 − 2.42i)14-s + (−0.5 − 0.866i)16-s + (2.43 − 4.22i)17-s + (−6.30 + 3.63i)19-s − 0.659·22-s + (3.98 − 2.29i)23-s + (3.06 + 5.31i)26-s + (2.12 + 1.57i)28-s + 8.09i·29-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.111 + 0.993i)7-s + 0.353i·8-s + (0.172 + 0.0993i)11-s − 1.70i·13-s + (−0.282 − 0.648i)14-s + (−0.125 − 0.216i)16-s + (0.591 − 1.02i)17-s + (−1.44 + 0.834i)19-s − 0.140·22-s + (0.830 − 0.479i)23-s + (0.601 + 1.04i)26-s + (0.402 + 0.296i)28-s + 1.50i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4232960123\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4232960123\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.295 - 2.62i)T \) |
good | 11 | \( 1 + (-0.570 - 0.329i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 6.13iT - 13T^{2} \) |
| 17 | \( 1 + (-2.43 + 4.22i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (6.30 - 3.63i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.98 + 2.29i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 8.09iT - 29T^{2} \) |
| 31 | \( 1 + (-0.759 - 0.438i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.05 + 8.75i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.25T + 41T^{2} \) |
| 43 | \( 1 + 9.03T + 43T^{2} \) |
| 47 | \( 1 + (-6.00 - 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (10.5 + 6.10i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.06 + 7.04i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.0618 - 0.0357i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.666 - 1.15i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.60iT - 71T^{2} \) |
| 73 | \( 1 + (2.44 + 1.41i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.88 + 5.00i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7.44T + 83T^{2} \) |
| 89 | \( 1 + (2.66 + 4.61i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 11.4iT - 97T^{2} \) |
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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
−8.457485978017494108577569329111, −7.86880508724888256674616447333, −6.97481380373459203076882031232, −6.25097906007067770610186735087, −5.40396730063998344474571808761, −4.96737902360356861960145303904, −3.44751482879048554965530372348, −2.71949394203775902153203276761, −1.57932205491744375333052614735, −0.16410630400230447268658499764,
1.28707100141906610248320099567, 2.12928458071290055951116132290, 3.40724678686088853342088150081, 4.12534373199700575686309201732, 4.82767651672071946812942265667, 6.28784012032760489314586102941, 6.71629263056520239680052705492, 7.41974970474554264149386414986, 8.401466029028478498959918675262, 8.815298495540326543516096131303