| 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
−12.96379091216685117716380565996, −11.65199795300480670407406180393, −10.67685823300876563960264732600, −10.03925335380853220448236259169, −9.170353994736542204337273714165, −7.968436230961955608531622409607, −6.44320398138846735275513964362, −4.90002416189013332764053586448, −4.09986363736027153361871332615, −2.70173880116177494260282941754,
1.82462406168472444305240798371, 2.60137900663216678147985517569, 5.73652924279862716931396883067, 6.29524223342366982276121166945, 6.94049554371557926055569626529, 8.246313059358244059030484351494, 9.348348038962713697162239175429, 10.95375941975333532635637166128, 11.53573146466162065532794445876, 12.68480101217960375273845361085
