| L(s) = 1 | + (−0.0484 − 0.924i)2-s + (−1.02 − 0.831i)3-s + (1.13 − 0.119i)4-s + (1.84 − 1.25i)5-s + (−0.718 + 0.989i)6-s + (−0.0993 + 2.64i)7-s + (−0.455 − 2.87i)8-s + (−0.260 − 1.22i)9-s + (−1.25 − 1.64i)10-s + (0.769 + 0.163i)11-s + (−1.26 − 0.822i)12-s + (0.371 + 0.729i)13-s + (2.44 − 0.0362i)14-s + (−2.94 − 0.246i)15-s + (−0.398 + 0.0847i)16-s + (−6.94 − 2.66i)17-s + ⋯ |
| L(s) = 1 | + (−0.0342 − 0.653i)2-s + (−0.592 − 0.479i)3-s + (0.568 − 0.0597i)4-s + (0.826 − 0.562i)5-s + (−0.293 + 0.403i)6-s + (−0.0375 + 0.999i)7-s + (−0.160 − 1.01i)8-s + (−0.0869 − 0.409i)9-s + (−0.395 − 0.521i)10-s + (0.232 + 0.0493i)11-s + (−0.365 − 0.237i)12-s + (0.103 + 0.202i)13-s + (0.654 − 0.00969i)14-s + (−0.760 − 0.0635i)15-s + (−0.0997 + 0.0211i)16-s + (−1.68 − 0.646i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0148 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0148 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.849712 - 0.862386i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.849712 - 0.862386i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.84 + 1.25i)T \) |
| 7 | \( 1 + (0.0993 - 2.64i)T \) |
| good | 2 | \( 1 + (0.0484 + 0.924i)T + (-1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (1.02 + 0.831i)T + (0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-0.769 - 0.163i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.371 - 0.729i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (6.94 + 2.66i)T + (12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (0.192 - 1.83i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-4.08 + 0.213i)T + (22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (-5.50 - 7.57i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.92 - 4.31i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (3.57 - 5.50i)T + (-15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-2.40 - 0.783i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-1.79 - 1.79i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.12 - 10.7i)T + (-34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (-5.66 + 6.99i)T + (-11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (7.62 + 8.47i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (6.28 + 5.65i)T + (6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-0.327 + 0.852i)T + (-49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (-0.767 + 0.557i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (3.11 - 2.02i)T + (29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (1.98 - 4.46i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (5.14 - 0.814i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-0.386 + 0.429i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-10.5 - 1.67i)T + (92.2 + 29.9i)T^{2} \) |
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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
−12.40781100221945329340000208613, −11.60967093574329420441843488750, −10.73725572398797317519044115373, −9.436069015751026739616509492544, −8.774967155365693065253922022492, −6.77898188238296511467750410131, −6.25763552061024778414481729543, −4.95760381580388710339361403548, −2.83458875586324885075724724857, −1.45747071133237516148420694300,
2.40143986158771154819200572592, 4.38911392962960659365578726995, 5.75848867397097441684860953935, 6.58410999475057386473252086257, 7.48883478554001017289980746042, 8.875662441571023114407142257150, 10.39308529380830637958260972740, 10.75544453930227723464551515822, 11.63220519411088682222038268850, 13.31879956671719686290218979547
