| L(s) = 1 | + (−0.142 − 0.319i)2-s + (−2.46 − 2.21i)3-s + (1.25 − 1.39i)4-s + (−1.42 − 1.72i)5-s + (−0.358 + 1.10i)6-s + (−0.494 + 2.59i)7-s + (−1.29 − 0.419i)8-s + (0.832 + 7.92i)9-s + (−0.347 + 0.701i)10-s + (0.207 − 1.97i)11-s + (−6.18 + 0.649i)12-s + (0.402 + 0.553i)13-s + (0.901 − 0.212i)14-s + (−0.299 + 7.39i)15-s + (−0.342 − 3.26i)16-s + (−0.429 − 2.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.226i)2-s + (−1.42 − 1.27i)3-s + (0.628 − 0.697i)4-s + (−0.638 − 0.769i)5-s + (−0.146 + 0.450i)6-s + (−0.186 + 0.982i)7-s + (−0.456 − 0.148i)8-s + (0.277 + 2.64i)9-s + (−0.109 + 0.221i)10-s + (0.0624 − 0.594i)11-s + (−1.78 + 0.187i)12-s + (0.111 + 0.153i)13-s + (0.240 − 0.0566i)14-s + (−0.0774 + 1.91i)15-s + (−0.0857 − 0.815i)16-s + (−0.104 − 0.489i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0320132 + 0.525455i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0320132 + 0.525455i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.42 + 1.72i)T \) |
| 7 | \( 1 + (0.494 - 2.59i)T \) |
| good | 2 | \( 1 + (0.142 + 0.319i)T + (-1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (2.46 + 2.21i)T + (0.313 + 2.98i)T^{2} \) |
| 11 | \( 1 + (-0.207 + 1.97i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-0.402 - 0.553i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.429 + 2.01i)T + (-15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (1.67 + 1.86i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (2.82 + 6.35i)T + (-15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (-0.118 - 0.365i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (4.05 - 0.861i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-4.85 + 0.510i)T + (36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (-6.19 + 4.49i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 2.62iT - 43T^{2} \) |
| 47 | \( 1 + (0.331 - 1.55i)T + (-42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-2.10 - 1.89i)T + (5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (2.16 + 0.963i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-0.0852 + 0.0379i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (1.96 + 9.26i)T + (-61.2 + 27.2i)T^{2} \) |
| 71 | \( 1 + (3.67 + 11.2i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-15.8 - 1.66i)T + (71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-10.4 - 2.22i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-14.3 - 4.67i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (11.4 - 5.08i)T + (59.5 - 66.1i)T^{2} \) |
| 97 | \( 1 + (-5.59 + 1.81i)T + (78.4 - 57.0i)T^{2} \) |
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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
−12.23254248257423024564819334245, −11.35358424002101357240808621687, −10.77670234152095960342763948625, −9.124071455851352325644099276277, −7.87903066528654661499662319413, −6.66051131653479545585201032760, −5.90032828197072600579909715117, −4.97300232721313984491412100940, −2.19107341182592240464967078937, −0.57575205135723874295776053163,
3.56478619906655577519432698064, 4.24662341674781319426363851304, 5.95605246968426657221807227472, 6.83648968987117357593127375109, 7.82796140022680177528403562317, 9.611031393695440973382317394624, 10.51738521439580070756070230416, 11.17247718539082718424664512203, 11.85102179149239333583438022113, 12.84810155488999565709209289622
