| L(s) = 1 | + (−1.84 + 2.05i)2-s + (0.933 − 0.358i)3-s + (−0.589 − 5.61i)4-s + (−0.166 + 2.22i)5-s + (−0.990 + 2.58i)6-s + (−3.35 − 1.93i)7-s + (8.14 + 5.92i)8-s + (0.743 − 0.669i)9-s + (−4.27 − 4.46i)10-s + (−0.431 + 0.0226i)11-s + (−2.56 − 5.02i)12-s + (1.56 − 3.24i)13-s + (10.1 − 3.30i)14-s + (0.643 + 2.14i)15-s + (−16.1 + 3.44i)16-s + (−1.57 + 4.10i)17-s + ⋯ |
| L(s) = 1 | + (−1.30 + 1.45i)2-s + (0.539 − 0.206i)3-s + (−0.294 − 2.80i)4-s + (−0.0745 + 0.997i)5-s + (−0.404 + 1.05i)6-s + (−1.26 − 0.731i)7-s + (2.88 + 2.09i)8-s + (0.247 − 0.223i)9-s + (−1.35 − 1.41i)10-s + (−0.130 + 0.00682i)11-s + (−0.739 − 1.45i)12-s + (0.433 − 0.901i)13-s + (2.72 − 0.884i)14-s + (0.166 + 0.552i)15-s + (−4.04 + 0.860i)16-s + (−0.381 + 0.995i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.420 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.420 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.400429 + 0.627059i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.400429 + 0.627059i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.933 + 0.358i)T \) |
| 5 | \( 1 + (0.166 - 2.22i)T \) |
| 13 | \( 1 + (-1.56 + 3.24i)T \) |
| good | 2 | \( 1 + (1.84 - 2.05i)T + (-0.209 - 1.98i)T^{2} \) |
| 7 | \( 1 + (3.35 + 1.93i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.431 - 0.0226i)T + (10.9 - 1.14i)T^{2} \) |
| 17 | \( 1 + (1.57 - 4.10i)T + (-12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (-6.03 - 2.31i)T + (14.1 + 12.7i)T^{2} \) |
| 23 | \( 1 + (2.29 - 0.120i)T + (22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (-2.17 + 4.88i)T + (-19.4 - 21.5i)T^{2} \) |
| 31 | \( 1 + (-8.18 + 1.29i)T + (29.4 - 9.57i)T^{2} \) |
| 37 | \( 1 + (-1.23 - 5.81i)T + (-33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (2.44 - 3.75i)T + (-16.6 - 37.4i)T^{2} \) |
| 43 | \( 1 + (-2.12 + 0.568i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (-6.59 - 9.07i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (0.338 - 2.13i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (0.749 + 0.0392i)T + (58.6 + 6.16i)T^{2} \) |
| 61 | \( 1 + (-9.15 - 1.94i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (0.826 - 7.86i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-2.57 - 2.08i)T + (14.7 + 69.4i)T^{2} \) |
| 73 | \( 1 + (-2.64 - 8.13i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.45 - 10.2i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (4.21 - 5.79i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (0.465 + 8.88i)T + (-88.5 + 9.30i)T^{2} \) |
| 97 | \( 1 + (1.12 + 10.7i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−10.13085492335573682334895353065, −9.475029345750209709904047686734, −8.222136030662024839079394807090, −7.86775814305229797031345208937, −6.98731149462712968492841332135, −6.35695952410241447203820956633, −5.77183647951540282507981332619, −4.06051807115505388029719535496, −2.77912738799454326557843988890, −0.988091892488883078337320639799,
0.65775387591066857778625489835, 2.09876780015307698464017475840, 3.01967780280756400381879780095, 3.86696423824855994421456591359, 5.00675397252885785929495509821, 6.72012090982610632110378857788, 7.66721481441258524322031092755, 8.686629414670578924412447219223, 9.104054660211005560232420877040, 9.533238708264292693144979948177
