L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + 5-s + (−2.36 + 1.19i)7-s + 0.999i·8-s + (−0.866 + 0.5i)10-s + 3.17i·11-s + (−1.82 + 1.05i)13-s + (1.44 − 2.21i)14-s + (−0.5 − 0.866i)16-s + (0.0900 + 0.155i)17-s + (−5.17 − 2.98i)19-s + (0.499 − 0.866i)20-s + (−1.58 − 2.74i)22-s − 0.789i·23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 0.447·5-s + (−0.892 + 0.451i)7-s + 0.353i·8-s + (−0.273 + 0.158i)10-s + 0.957i·11-s + (−0.505 + 0.292i)13-s + (0.386 − 0.592i)14-s + (−0.125 − 0.216i)16-s + (0.0218 + 0.0378i)17-s + (−1.18 − 0.685i)19-s + (0.111 − 0.193i)20-s + (−0.338 − 0.586i)22-s − 0.164i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2282829722\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2282829722\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (2.36 - 1.19i)T \) |
good | 11 | \( 1 - 3.17iT - 11T^{2} \) |
| 13 | \( 1 + (1.82 - 1.05i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.0900 - 0.155i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.17 + 2.98i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 0.789iT - 23T^{2} \) |
| 29 | \( 1 + (-6.84 - 3.95i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (8.40 + 4.85i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.27 + 7.39i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.85 + 10.1i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.84 - 3.20i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.04 - 1.80i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.613 - 0.353i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.88 + 10.1i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.50 + 1.44i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.97 - 8.62i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.1iT - 71T^{2} \) |
| 73 | \( 1 + (5.09 - 2.93i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.50 + 9.53i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.41 - 7.64i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.91 - 5.04i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.79 + 2.19i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.115587457955340845001981186340, −8.337613719383155919551006162576, −7.16633764764304032478985482210, −6.80296887603173563255350421966, −5.92199397957775901245667721257, −5.08294353859473247225886809125, −4.05555151526012115822887901919, −2.64903556696451063289819039425, −1.92031226492777197683568463138, −0.10222691026720897099966530611,
1.27629843031334059219683222190, 2.62755439497781679331843037022, 3.37381176963141787175347864620, 4.40150107949396537523550708638, 5.65751748069858073260017252649, 6.41484724394953346007323533889, 7.08153803466211351968457262563, 8.166851300260158827836630432035, 8.664410918143937451433125344180, 9.635344705522532237053172876380