L(s) = 1 | − i·2-s − 4-s + i·5-s + i·8-s + 10-s − 6i·11-s + (2 + 3i)13-s + 16-s + 6·17-s + 6i·19-s − i·20-s − 6·22-s − 6·23-s − 25-s + (3 − 2i)26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.447i·5-s + 0.353i·8-s + 0.316·10-s − 1.80i·11-s + (0.554 + 0.832i)13-s + 0.250·16-s + 1.45·17-s + 1.37i·19-s − 0.223i·20-s − 1.27·22-s − 1.25·23-s − 0.200·25-s + (0.588 − 0.392i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.596202411\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.596202411\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-2 - 3i)T \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 12iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−9.907061829539561390545471456008, −8.765396101768882876123171316971, −8.279392344904790297936886066097, −7.27506480875146797858663194066, −5.91144538378184232459021474295, −5.69704405657879966704216041612, −3.93127204330606369456829698447, −3.55936500404090664176720540899, −2.30563602120247477932610062637, −0.926595062772527519945005106874,
1.08113414596064056073134167321, 2.67556638975386062917203189118, 4.07177599827163527303779402571, 4.84947260935169233541942111350, 5.65912914489575322981076212751, 6.63336714778307000007434443473, 7.51457246243163340450269475596, 8.101352568933873853463339446835, 9.004705083439948069743582414396, 9.953279093252913134450652125731