| L(s) = 1 | + (0.107 + 0.278i)2-s + (0.163 − 3.11i)3-s + (1.42 − 1.27i)4-s + (1.71 + 1.43i)5-s + (0.887 − 0.288i)6-s + (−0.510 + 2.59i)7-s + (1.04 + 0.530i)8-s + (−6.71 − 0.706i)9-s + (−0.215 + 0.632i)10-s + (−0.245 − 2.33i)11-s + (−3.75 − 4.63i)12-s + (0.415 + 2.62i)13-s + (−0.778 + 0.135i)14-s + (4.74 − 5.12i)15-s + (0.363 − 3.45i)16-s + (−6.25 + 4.06i)17-s + ⋯ |
| L(s) = 1 | + (0.0756 + 0.197i)2-s + (0.0943 − 1.80i)3-s + (0.710 − 0.639i)4-s + (0.768 + 0.640i)5-s + (0.362 − 0.117i)6-s + (−0.192 + 0.981i)7-s + (0.367 + 0.187i)8-s + (−2.23 − 0.235i)9-s + (−0.0680 + 0.199i)10-s + (−0.0739 − 0.703i)11-s + (−1.08 − 1.33i)12-s + (0.115 + 0.727i)13-s + (−0.208 + 0.0362i)14-s + (1.22 − 1.32i)15-s + (0.0907 − 0.863i)16-s + (−1.51 + 0.985i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.433 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24584 - 0.782819i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24584 - 0.782819i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.71 - 1.43i)T \) |
| 7 | \( 1 + (0.510 - 2.59i)T \) |
| good | 2 | \( 1 + (-0.107 - 0.278i)T + (-1.48 + 1.33i)T^{2} \) |
| 3 | \( 1 + (-0.163 + 3.11i)T + (-2.98 - 0.313i)T^{2} \) |
| 11 | \( 1 + (0.245 + 2.33i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (-0.415 - 2.62i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (6.25 - 4.06i)T + (6.91 - 15.5i)T^{2} \) |
| 19 | \( 1 + (-0.733 + 0.815i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (2.50 - 0.963i)T + (17.0 - 15.3i)T^{2} \) |
| 29 | \( 1 + (-1.76 - 0.573i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.156 + 0.734i)T + (-28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-7.58 + 6.13i)T + (7.69 - 36.1i)T^{2} \) |
| 41 | \( 1 + (2.68 - 3.69i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-5.10 + 5.10i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.35 - 2.09i)T + (-19.1 - 42.9i)T^{2} \) |
| 53 | \( 1 + (6.99 + 0.366i)T + (52.7 + 5.54i)T^{2} \) |
| 59 | \( 1 + (-5.83 + 2.59i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-0.649 + 1.45i)T + (-40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-1.30 - 2.00i)T + (-27.2 + 61.2i)T^{2} \) |
| 71 | \( 1 + (-0.0626 + 0.192i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.83 - 2.26i)T + (-15.1 - 71.4i)T^{2} \) |
| 79 | \( 1 + (-0.592 - 2.78i)T + (-72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (0.288 - 0.565i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-11.0 - 4.91i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (3.52 + 6.92i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−12.68480101217960375273845361085, −11.53573146466162065532794445876, −10.95375941975333532635637166128, −9.348348038962713697162239175429, −8.246313059358244059030484351494, −6.94049554371557926055569626529, −6.29524223342366982276121166945, −5.73652924279862716931396883067, −2.60137900663216678147985517569, −1.82462406168472444305240798371,
2.70173880116177494260282941754, 4.09986363736027153361871332615, 4.90002416189013332764053586448, 6.44320398138846735275513964362, 7.968436230961955608531622409607, 9.170353994736542204337273714165, 10.03925335380853220448236259169, 10.67685823300876563960264732600, 11.65199795300480670407406180393, 12.96379091216685117716380565996
