L(s) = 1 | + 2.84i·3-s − 5-s + (−0.712 + 2.54i)7-s − 5.08·9-s − 5.41·11-s + 0.641·13-s − 2.84i·15-s + 5.16i·17-s + 3.44i·19-s + (−7.24 − 2.02i)21-s + 7.94i·23-s + 25-s − 5.94i·27-s − 10.0i·29-s + 5.38·31-s + ⋯ |
L(s) = 1 | + 1.64i·3-s − 0.447·5-s + (−0.269 + 0.963i)7-s − 1.69·9-s − 1.63·11-s + 0.177·13-s − 0.734i·15-s + 1.25i·17-s + 0.790i·19-s + (−1.58 − 0.442i)21-s + 1.65i·23-s + 0.200·25-s − 1.14i·27-s − 1.86i·29-s + 0.966·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0108 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0108 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5072608845\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5072608845\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (0.712 - 2.54i)T \) |
good | 3 | \( 1 - 2.84iT - 3T^{2} \) |
| 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 - 0.641T + 13T^{2} \) |
| 17 | \( 1 - 5.16iT - 17T^{2} \) |
| 19 | \( 1 - 3.44iT - 19T^{2} \) |
| 23 | \( 1 - 7.94iT - 23T^{2} \) |
| 29 | \( 1 + 10.0iT - 29T^{2} \) |
| 31 | \( 1 - 5.38T + 31T^{2} \) |
| 37 | \( 1 - 0.931iT - 37T^{2} \) |
| 41 | \( 1 + 8.33iT - 41T^{2} \) |
| 43 | \( 1 + 6.76T + 43T^{2} \) |
| 47 | \( 1 - 8.86T + 47T^{2} \) |
| 53 | \( 1 - 6.31iT - 53T^{2} \) |
| 59 | \( 1 + 11.5iT - 59T^{2} \) |
| 61 | \( 1 - 0.834T + 61T^{2} \) |
| 67 | \( 1 + 2.01T + 67T^{2} \) |
| 71 | \( 1 - 0.628iT - 71T^{2} \) |
| 73 | \( 1 - 10.3iT - 73T^{2} \) |
| 79 | \( 1 + 2.98iT - 79T^{2} \) |
| 83 | \( 1 + 11.2iT - 83T^{2} \) |
| 89 | \( 1 + 13.3iT - 89T^{2} \) |
| 97 | \( 1 - 9.07iT - 97T^{2} \) |
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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
−9.862340052199966568194670993734, −8.853084505438688010700870720338, −8.257333406950046958174494659076, −7.61038257771558316259316318918, −6.02959231609745097397731336280, −5.62192143809436693277665641451, −4.81905214375518929022173785706, −3.89703323671146191454950755804, −3.24428630734616561976454734882, −2.22457243425545873051792133504,
0.19786460350329547695592931922, 1.04406486672315284661363169514, 2.53675800359527101060742319959, 3.04344404820361631122910962950, 4.55656258551274126837968026400, 5.28963916577635910804629127438, 6.54246438293579499179795061471, 6.94260181684950600562795945277, 7.60348970237964786344777645594, 8.172302147108561998009396223801