| L(s) = 1 | + (0.0490 − 0.466i)2-s + (−2.53 − 0.537i)3-s + (1.74 + 0.370i)4-s + (1.74 + 1.39i)5-s + (−0.374 + 1.15i)6-s + (−1.07 + 2.41i)7-s + (0.547 − 1.68i)8-s + (3.37 + 1.50i)9-s + (0.736 − 0.745i)10-s + (3.90 − 1.73i)11-s + (−4.20 − 1.87i)12-s + (3.34 − 2.43i)13-s + (1.07 + 0.617i)14-s + (−3.66 − 4.47i)15-s + (2.49 + 1.11i)16-s + (1.53 + 1.70i)17-s + ⋯ |
| L(s) = 1 | + (0.0346 − 0.329i)2-s + (−1.46 − 0.310i)3-s + (0.870 + 0.185i)4-s + (0.781 + 0.624i)5-s + (−0.153 + 0.470i)6-s + (−0.404 + 0.914i)7-s + (0.193 − 0.595i)8-s + (1.12 + 0.500i)9-s + (0.232 − 0.235i)10-s + (1.17 − 0.523i)11-s + (−1.21 − 0.540i)12-s + (0.928 − 0.674i)13-s + (0.287 + 0.165i)14-s + (−0.947 − 1.15i)15-s + (0.623 + 0.277i)16-s + (0.371 + 0.412i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05526 - 0.0365820i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05526 - 0.0365820i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.74 - 1.39i)T \) |
| 7 | \( 1 + (1.07 - 2.41i)T \) |
| good | 2 | \( 1 + (-0.0490 + 0.466i)T + (-1.95 - 0.415i)T^{2} \) |
| 3 | \( 1 + (2.53 + 0.537i)T + (2.74 + 1.22i)T^{2} \) |
| 11 | \( 1 + (-3.90 + 1.73i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (-3.34 + 2.43i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.53 - 1.70i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (6.32 - 1.34i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (0.132 - 1.26i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (0.0837 + 0.257i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (3.38 + 3.76i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (9.27 + 4.12i)T + (24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (3.17 - 2.30i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 1.64T + 43T^{2} \) |
| 47 | \( 1 + (-1.32 + 1.47i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (-7.56 - 1.60i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.431 - 4.10i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-0.0371 + 0.353i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (4.42 + 4.91i)T + (-7.00 + 66.6i)T^{2} \) |
| 71 | \( 1 + (4.19 + 12.8i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.642 - 0.286i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (10.9 - 12.1i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-3.36 + 10.3i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-0.893 + 8.49i)T + (-87.0 - 18.5i)T^{2} \) |
| 97 | \( 1 + (4.91 + 15.1i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−12.43595406043794222646773220708, −11.66671778250964758291474269026, −10.88610973724051419226271212720, −10.22739321322783006022977173565, −8.741577122345515665587446269747, −6.98650383058790922187528263579, −6.04398951014101881717234419102, −5.87320088547603447789670852740, −3.47558983162010856757952048764, −1.75009083042409293631088789617,
1.45234365418791460897164068565, 4.19288075517884650908653857618, 5.38826616444345202708845645411, 6.54574251784901435045756299199, 6.78219714535447111356400028612, 8.795081457898317588700665093617, 10.09447221864862872560263437272, 10.72132805702862855962613890659, 11.68829199699047235138325752079, 12.46927903177092935795426590752
